GENERALISED HEEGNER CYCLES AND p-ADIC RANKIN L-SERIES

MASSIMO BERTOLINI, HENRI DARMON & KARTIK PRASANNA

1

(with an appendix by BRIAN CONRAD^{2})

Contents

Introduction 2 1. Preliminaries 6 1.1. Algebraic modular forms 6 1.2. Modular forms over C 9 1.3. p-adic modular forms 11 1.4. Elliptic curves with complex multiplication 12 1.5. Values of modular forms at CM points 14 2. Generalised Heegner cycles 15 2.1. Kuga-Sato varieties 15 2.2. The variety X_{r} and its cohomology 18 2.3. Definition of the cycles 19 2.4. Relation with Heegner cycles and L-series 19 3. p-adic Abel-Jacobi maps 20 3.1. The étale Abel-Jacobi map 20 3.2. The comparison isomorphism 21 3.3. Extensions of filtered Frobenius modules 22 3.4. The p-adic Abel-Jacobi map 23 3.5. de Rham cohomology over p-adic fields 24 3.6. The Coleman primitive 27 3.7. p-adic integration and the p-adic Abel-Jacobi map 28 3.8. Calculation of the Coleman primitive 31 4. Period integrals and central values of Rankin-Selberg L-functions 33 4.1. Rankin L-series and their special values 33 4.2. Differential operators 38 4.3. Period integrals and values at CM points 39 4.4. Explicit theta lifts 42 4.5. Seesaw duality and the Siegel-Weil formula 49 4.6. Local zeta integrals 51 4.7. The explicit form of Waldspurger’s formula 56 5. Anticyclotomic p-adic L-functions 56 5.1. Periods and algebraicity 56 5.2. p-adic interpolation 60 5.3. The main theorem 62 Appendix A. Kuga-Sato schemes

by Brian Conrad 64

References 67

1Supported partially by NSF grants DMS-1015173 and DMS-0854900.

2Supported partially by NSF grant DMS-1100784.

Introduction

This article studies a distinguished collection of algebraic cycles on varieties which are fibered over modular curves. The cycles in question generalise the Heegner cycles on Kuga-Sato varieties that are studied in [Sch], [Ne2] and [Zh], and will henceforth be referred to as generalised Heegner cycles. The main result (Theorem 5.13 of Chapter 5.3) is a p-adic analogue of the Gross-Zagier formula which relates the images of generalised Heegner cycles under a p-adic Abel-Jacobi map to the special values of certain p-adic Rankin L-series at critical points that lie outside the range of p-adic interpolation. Even in the 0-dimensional limit case where generalised Heegner cycles are nothing but Heegner divisors on modular curves, this analogue differs from the p-adic Gross-Zagier formula proved in [PR1], and provides a concrete instance of the p-adic Beilinson conjectures of [PR2], [PR3]. It can also be viewed as the direct analogue of Leopoldt’s evaluation at s = 1 of the classical p-adic L-function attached to an even Dirichlet character in terms of p-adic logarithms of cyclotomic units. In this analogy, the Kubota-Leopoldt p-adic L-function is replaced by the p-adic Rankin L-function attached to a cusp form and a theta series of an imaginary quadratic field, and the cyclotomic units are replaced by (generalised) Heegner cycles.

Recall that the Kuga-Sato variety W_{r} is a smooth compactification of the r-fold product of the universal generalised elliptic curve over a modular curve C = C_{Γ} attached to Γ = Γ_{1}(N). It is naturally fibered over C, with generic fiber isomorphic to an r-fold product of elliptic curves. The variety W_{2r} is equipped with a supply of so-called Heegner cycles (in the Chow group with rational coefficients) of dimension r, which were introduced in [GZ], §V.4. (See also [Ne2], §II.3.6, where a more precise definition is given.) These cycles are supported on fibers above CM points of C and are defined over abelian extensions of imaginary quadratic fields. The main theorem of [Zh] relates their heights to the central critical derivatives of Rankin convolution L-series of cusp forms of weight 2r + 2 with weight one binary theta series attached to finite order Hecke characters of an imaginary quadratic field. In the case r = 0, where the Heegner cycles are Heegner points on the modular curve C = W_{0}, this is the theorem of Gross and Zagier [GZ]. A p-adic analogue of these formulae has also been established (in [PR1] for r = 0 and in [Ne2] for general r) in which the Arakelov height pairing is replaced by a p-adic height pairing and the complex L-series by a suitable two-variable p-adic L-function.

The present work replaces the Kuga-Sato variety W_{2r} by the (2r+1)-dimensional variety

X_{r} := W_{r} × A^{r},

where A is a fixed elliptic curve with complex multiplication by the ring of integers of an imaginary quadratic field K, defined, say, over the Hilbert class field H of K. Like W_{2r}, the variety X_{r} is fibered over the modular curve C and is also equipped with an infinite collection of special cycles defined over abelian extensions of K. These generalised Heegner cycles are naturally indexed by isogenies φ : A⟶A′. The cycle attached to φ, denoted Δ_{φ}, is supported on the fiber (A′)^{r} × A^{r} above a point of C attached to A′, and is essentially equal to the r-fold self-product of the graph of φ.

Section 2.3 of Chapter 2 defines the cycles Δ_{φ} precisely and establishes some of their basic properties. In particular, it shows that generalised Heegner cycles are homologically trivial. One can therefore consider their images under various (étale, p-adic, and also complex) Abel-Jacobi maps defined on homologically trivial cycles modulo rational equivalence. Moreover, it is observed in Section 2.4 that the classical Heegner cycles on W_{2r} attached to the imaginary quadratic field K can be obtained as the images of generalised Heegner cycles on X_{2r} under a suitable algebraic correspondence. It follows that generalised Heegner cycles carry at least as much arithmetic information as Heegner cycles on Kuga-Sato varieties. One expects that they carry substantially more: namely, that their heights should encode the central critical derivatives of Rankin L-series attached to the convolution of cusp forms of weight k := r + 2 on Γ with theta series of weight ≤ k − 1 attached to certain Hecke characters of K (and not just with those arising from finite order characters).

Chapter 3 describes the images of generalised Heegner cycles under the p-adic Abel-Jacobi map for a prime p not dividing N. More precisely, Section 3.1 introduces the étale Abel-Jacobi map

(0.0.1)  AJ_{F}^{et} : CH^{r+1}(X_{r})_{0,Q}(F)⟶H^{1}(F, H_{et}^{2r+1}(X̄_{r}, Q_{p})(r+1))

attached to any field F containing H, where H^{1}(F, M) denotes the (continuous) group cohomology of G_{F} := Gal(F̄/F) with values in a G_{F} –module M. (Here and elsewhere, the subscript 0 stands for homologically trivial and the subscript Q denotes the Chow group with rational coefficients.) As shown in Appendix A, the variety X_{r} admits a proper smooth model over Spec Z[\frac{1}{N}] and hence the image of AJ_{F}^{et} (for F a finite extension of Q_{p}) is contained in the Bloch-Kato subspace H_{f}^{1}. The comparison theorems between p-adic étale cohomology and de Rham cohomology then allow us to view (0.0.1) as a map AJ_{F} (called the p-adic Abel-Jacobi map)

(0.0.2)  AJ_{F} : CH^{r+1}(X_{r})_{0,Q}(F)⟶ Fil^{r+1} H_{dR}^{2r+1}(X_{r}/F)^{∨}.

Chapter 3 explains how this map can be computed analytically via Coleman’s theory of p-adic integration of differential forms attached to certain classes in the de Rham cohomology H_{dR}^{2r+1}(X_{r}/F).

We now describe briefly the anticyclotomic p-adic L-function that is constructed in Chapters 4 and 5. Let S_{k}(Γ_{0}(N), ε) denote the space of cusp forms of weight k, level N and character ε. The quadratic imaginary field K is said to satisfy the Heegner hypothesis (relative to N) if O_{K} possesses a cyclic ideal 𝔑 of norm N, i.e., an ideal for which

(0.0.3)  O_{K}/𝔑 = Z/NZ.

Assume that this hypothesis is satisfied, and fix a normalised newform f ∈ S_{k}(Γ_{0}(N), ε_{f}). Let χ be a Hecke character of K of infinity type (j_{1}, j_{2}) with j_{1} + j_{2} = k and satisfying

(0.0.4)  χ|_{𝔸_{Q}^{×}} = ε_{f} · N^{k}

where N is the usual norm character. This condition implies that the Rankin L-series L(f, χ^{−1}, s) is selfdual and its functional equation relates its values at s to those at −s, so that 0 is the point of symmetry. Such χ will be called central critical for f.

At the cost of possibly interchanging j_{1} and j_{2}, we will assume that j_{1} ≥ 0. Let Σ_{cc}(𝔑) denote the set of central critical characters of conductor dividing 𝔑 and satisfying (0.0.4), as well as the following auxiliary condition: for all finite primes q, the epsilon factor ε_{q}(f, χ^{−1}) = +1. Given our other hypotheses, this auxiliary condition is automatic except at those primes q ramified in K, that divide N but do not divide the conductor of ε_{f}. (In the text, we allow more generally the conductor of χ to divide c𝔑 where c is an auxiliary odd rational integer prime to Nd_{K}, where −d_{K} is the discriminant of K.) The set Σ_{cc}(𝔑) can be written as the disjoint union of two subsets:

Σ_{cc}(𝔑) = Σ_{cc}^{(1)}(𝔑) ∪ Σ_{cc}^{(2)}(𝔑),

where Σ_{cc}^{(1)}(𝔑) consists of the characters of infinity type (k−1−j, 1+j) with 0 ≤ j ≤ r, and Σ_{cc}^{(2)}(𝔑) consists of those of infinity type (k + j, −j) with j ≥ 0. When χ ∈ Σ_{cc}^{(1)}(𝔑), the sign ε_{∞}(f, χ^{−1}) equals −1, hence the sign in the functional equation for L(f, χ^{−1}, s) is also −1, and therefore the function χ ↦ L(f, χ^{−1}, 0) vanishes identically on Σ_{cc}^{(1)}(𝔑). On the other hand, for χ ∈ Σ_{cc}^{(2)}(𝔑), the sign ε_{∞}(f, χ^{−1}) equals +1 whence the sign in the functional equation for L(f, χ^{−1}, s) is +1 as well, and so one expects that the associated central critical values should be non-zero most of the time.

Chapter 4 is devoted to proving an explicit version of Waldspurger’s formula relating the central Lvalues L(f, χ^{−1}, 0), for χ ∈ Σ_{cc}^{(2)}(𝔑), to period integrals on tori. Such explicit formulae have been studied by several authors recently, for example [Xue], [MW] and more recently [Hi3]. However our approach is somewhat different in that we always insist that our torus embeddings come from Heegner points and that the test vectors are of minimal level. The relevant period integrals then reduce to finite sums of values of (certain non-holomorphic derivatives of) the form f at all conjugates of a CM point, twisted by the character χ^{−1}, which is a key to providing a link to the p-adic Abel-Jacobi images of generalized Heegner cycles supported on the same set of conjugate CM points.

Section 5.1 recalls the algebraicity properties of these special values: for all χ ∈ Σ_{cc}^{(2)}(𝔑),

(0.0.5)  L_{alg}(f, χ^{−1}) := C̃(f, χ) × \frac{L(f, χ^{−1}, 0)}{Ω^{2(k+2j)}}

is an algebraic number. Here C̃(f, χ) is an explicit, elementary constant and Ω is a CM period attached to K whose value depends on the choice of a regular differential ω_{A} on A/H. After fixing an embedding

ι : Q̄⟶Q̄_{p},

the values L_{alg}(f, χ^{−1}) attached to χ ∈ Σ_{cc}^{(2)}(𝔑) can be viewed as p-adic numbers. Section 5.2 takes up the question of their p-adic interpolation. As explained in that section, the set Σ_{cc}^{(2)}(𝔑) is endowed with a natural p-adic topology, and can be viewed as a dense subset of its completion Σ̂_{cc}(𝔑). Assume that the rational prime p is split in K/Q, so that ι(K) ⊂ Q_{p}. Under this assumption, there is a unique prime 𝔭 of K above p for which χ(𝔭) is a p-adic unit. The main result of Section 5.2 is that, after setting

L_{p}(f, χ) = Ω_{p}^{2(k+2j)}(1 − χ^{−1}(𝔭̄)a_{p} + ε_{f}(p)χ(𝔭̄)^{−2}p^{k−1})^{2}L_{alg}(f, χ^{−1})

for an appropriate p-adic period Ω_{p} (which also depends on the choice of ω_{A}), the assignment χ ↦ L_{p}(f, χ^{−1}) extends to a (necessarily unique) continuous function on Σ̂_{cc}(𝔑), which we refer to as the anticyclotomic p-adic L-function attached to f and K.

Now, let χ be a character in Σ_{cc}^{(1)}(𝔑), having infinity type (k − 1 − j, 1 + j) for some 0 ≤ j ≤ r. While the classical L-value L(f, χ^{−1}, 0) vanishes, the character χ can be viewed as an element of Σ̂_{cc}(𝔑) (lying outside the range of classical interpolation defining the anticyclotomic p-adic L-function L_{p}(f, χ)), and the special value L_{p}(f, χ)—which may be thought of as a p-adic avatar of L′(f, χ^{−1}, 0)—is not forced to vanish a priori. Our main result relates L_{p}(f, χ) to the Abel-Jacobi images of generalised Heegner cycles. For the sake of illustration, we state the main result under the following simplifying assumptions, postponing the more general statement to Theorem 5.13 of Chapter 5.3:

(1) The quadratic imaginary field K has class number one and odd discriminant −d_{K} < −3. Let ε_{K} : (Z/d_{K}Z)^{×}⟶{±1} be the associated odd Dirichlet character, and denote by the same symbol the quadratic character of (O_{K}/\sqrt{−d_{K}}O_{K})^{×} induced from the identification of O_{K}/\sqrt{−d_{K}}O_{K} with Z/d_{K}Z.

(2) The newform f belongs to S_{k}(Γ_{0}(N), ε_{K}^{k}). (Note that it is necessary that d_{K} divides N when k is odd.)

(3) The grossencharacter χ ∈ Σ_{cc}^{(1)}(𝔑) is of the form

χ((α)) = ε_{K}^{k}(α)α^{k−1−j}ᾱ^{1+j},

for some integer 0 ≤ j ≤ r. In this special setting, our main result is:

Main Theorem. Let Δ = Δ_{1} be the generalised Heegner cycle attached to 1 : A⟶A, viewed as an element of CH^{r+1}(X_{r})_{0}(Q_{p})_{Q} via ι. Then

\frac{L_{p}(f, χ)}{Ω_{p}^{2(r−2j)}} = (1 − χ^{−1}(𝔭̄)a_{p} + χ^{−2}(𝔭̄)p^{k−1})^{2} · (\frac{1}{j!} AJ_{Q_{p}}(Δ)(ω_{f} ∧ ω_{A}^{j}η_{A}^{r−j}))^{2} ,

where AJ_{Q_{p}} is the p-adic Abel-Jacobi map of (0.0.2), ω_{f} is the class in Ω^{r+1}(W_{r}) attached to f in Corollary 2.3 of Section 2.1, and ω_{A}^{j}η_{A}^{r−j} is the class in H^{r}(A^{r}) defined in (1.4.6) of Section 1.4.

Note that it is a special value and not a derivative of the p-adic L-series that occurs on the analytic side of this formula, while the algebraic side involves the Abel-Jacobi images of generalised Heegner cycles rather than their (p-adic) heights. Note also that if ω_{A} is replaced by a nonzero multiple λω_{A}, then both sides of the equation above are multiplied by λ^{2(2j−r)}.

Those approaching this paper for the first time may find it pedagogically helpful to focus on the simplest case r = j = 0, where f is a newform of weight 2 and χ ∈ Σ_{cc}^{(1)}(𝔑) is a grossencharacter of infinity type (1, 1). In this case, the Main Theorem above involves the formal group logarithms of points in the Jacobians of modular curves arising from certain divisors supported on Heegner points. It relates these p-adic logarithms to the values of the p-adic L-function L_{p}(f, χ) at characters of finite order (shifted by the norm). One thus obtains a new p-adic variant of the Gross-Zagier formula in the “traditional” setting of Heegner points on modular curves. As a first guide to the somewhat lengthy arguments required to deal with forms and Hecke characters of general weights and levels, here is a brief outline of the proof of the Main Theorem in this simplest non-trivial setting, assuming further that K has class number one and a unit group of order 2, and that χ := χ_{0} is the trivial character of weight (1, 1) sending the (principal) ideal (α) to its norm αᾱ. This norm character is the specialisation at j = 0 of the sequence χ_{j} ∈ Σ_{cc}^{(2)}(𝔑) of grossencharacters of infinity type (1 + j, 1 − j) defined by

χ_{j}((α)) := α^{1+j}ᾱ^{1−j}.

Let δ_{2}^{j−1} denote the (j − 1)st iterate of the Shimura-Maass differential operator as defined in Sec. 1.2; this sends weight 2 real analytic modular forms to those of weight 2j. For all j ≥ 1, Theorem 5.5 identifies the quantity L_{alg}(f, χ_{j}^{−1}) of equation (0.0.5) with (δ_{2}^{j−1}f)(P_{A})^{2}, where P_{A} denotes the triple (A, ω_{A}, t_{A}) attached to the elliptic curve A with CM by the maximal order of K, the differential ω_{A} and a suitable Γ_{1}(N)-level structure t_{A} on A. (Here modular forms are viewed as functions on triples as explained in Section 1.1.) Using the well-known fact that the unit root splitting of the Hodge filtration agrees with the Hodge decomposition for ordinary CM elliptic curves, Proposition 1.12 identifies (δ_{2}^{j−1}f)(P_{A}) with (θ^{j−1}f)(P_{A}), where θ = q \frac{d}{dq} is the Atkin-Serre theta operator on p-adic modular forms defined in (1.3.2). This key identification leads to the p-adic interpolation of the special values L_{alg}(f, χ_{j}^{−1}) described in Section 5.2, and hence, to the Rankin p-adic L-function L_{p}(f, χ_{j}) which arises in the Main Theorem above. This p-adic L-function satisfies the equality

L_{p}(f, χ_{j}) = (θ^{j−1}f^{♭})(P_{A}^{(p)})^{2}, ∀j ≥ 0,

where f^{♭} is the “p-depleted” modular form associated to f defined in (3.8.4), and P_{A}^{(p)} = (A, Ω_{p}^{−1}ω_{A}, t_{A}). Taking a p-adic limit when j → 0 shows that

L_{p}(f, χ) = (θ^{−1}f^{♭})(P_{A}^{(p)})^{2}.

One can see (either directly, or by specializing the calculations of Section 3 to the case where r = 0) that the function θ^{−1}f^{♭} – a p-adic, and in fact overconvergent, modular form of weight 0 – is the unique rigid analytic primitive of the exact rigid differential ω_{f^{♭}} which vanishes at the cusp ∞, and its value at the triple P_{A}^{(p)} is an explicit multiple of the formal group logarithm, relative to the differential ω_{f}, of the degree zero divisor Δ_{1} = (A, t_{A}) − (∞) on the modular curve C.

We close this introduction by listing a few of the arithmetic applications of Theorem 5.13.

Rubin’s formula. The article [BDP-cm] exploits Theorem 5.13 in the special case where f is itself a weight two binary theta series attached to the quadratic imaginary field K to give a new proof of the main result of [Ru], which relates the values of the Katz p-adic L-function attached to K to the p-adic logarithms of global points on elliptic curves with complex multiplication by K.

Chow-Heegner points. Because it involves Abel-Jacobi images rather than p-adic heights, Theorem 5.13 is used in [BDP-ch] to study the algebraicity of the certain points on CM elliptic curves arising from higher dimensional cycles in the Chow groups of certain algebraic varieties whose cohomology realises the ℓ-adic repreentations attached to theta series of higher (possibly odd) weight. This construction, provides a basic illustration of the phenomenon of “Chow-Heegner points” arising from the image of algebraic cycles under Abel-Jacobi maps (both complex and p-adic). The relevance of Theorem 5.13 to the notion of Chow-Heegner points was in fact the original motivation for the present article, although Theorem 5.13 is considerably more general than the special case exploited in [BDP-ch].

Coniveau and the Bloch-Beilinson conjecture. The article [BDP-co] illustrates how Theorem 5.13 may be used to prove part of the Bloch-Beilinson conjecture for the Rankin-Selberg motives that are studied in this article. In particular, by verifying that specific values of the p-adic L-function L_{p}(f, χ) are not zero, one can often show that generalised Heegner cycles are not just nonzero in the Chow group but also nonzero in a certain graded piece for the coniveau filtration on the Chow group, as predicted by a refined version ([Bl-1], [Bl-2]) of the Bloch-Beilinson conjecture.

Euler systems. Let F be any global field over which A is defined. For each cuspidal newform f on C of weight r + 2 and each character χ as in the previous statement, there is a G_{F} –equivariant projection

π_{f,χ} : H_{et}^{2r+1}(X̄_{r}, Q_{p})(r+1)⟶(V_{f} ⊗ χ)(r+1) =: V_{f,χ},

where V_{f} is the Deligne representation attached to f and χ is viewed as a one-dimensional p-adic representation of G_{F} in the usual way. Each generalised Heegner cycle Δ_{φ}, defined over an appropriate extension F_{φ} ⊃ H, gives rise to a global cohomology class:

κ_{φ} := π_{f,χ}(AJ_{F_{φ}}^{et}(Δ_{φ})) ∈ H^{1}(F_{φ}, V_{f,χ}),

which belongs to a generalised Selmer group H_{Sel}^{1}(F_{φ}, V_{f,χ}) attached to the p-adic Galois representation V_{f,χ}. If 𝔭 is a prime of F_{φ} above p and p does not divide the level of Γ, the discriminant of K, or the degree of φ, then the natural image res_{𝔭}(κ_{φ}) of κ_{φ} in the local cohomology group H^{1}(F_{φ,𝔭}, V_{f,χ}) belongs to the subgroup H_{f}^{1}(F_{φ,𝔭}, V_{f,χ}) corresponding to cristalline extensions of V_{f,χ} by Q_{p}. The Main Theorem above relates res_{𝔭}(κ_{φ}) to the values of the p-adic L-function L_{p}(f, χ) at points lying outside the range of classical interpolation. This suggests that the collection {κ_{φ}} of global cohomology classes, as φ ranges over the isogenies A⟶A′, should give rise to an Euler system attached to the compatible system V_{f,χ} of p-adic representations of G_{F}. See Section 2.4 for a discussion of the relation between these cycles and classical L-series, and [Cas1], [Cas2] where the connection between the results of this paper and the theory of Euler systems obtained by interpolating generalised Heegner cycles in p-adic families is described in more detail.

Acknowledgements: The authors thank Fabrizio Andreatta and the four anonymous referees for constructive criticism which led to significant improvements in the organization and presentation of their results. In particular, the appendix by Brian Conrad was added after a gap in the literature was brought to their attention. The authors are also grateful to Brian Conrad for kindly agreeing to supply this appendix and for pointing out several corrections to a previous version of this work.

1. Preliminaries

1.1. Algebraic modular forms. Let N ≥ 1 be an integer and let Γ = Γ_{1}(N) be the standard congruence subgroup of level N:

Γ_{1}(N) := \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(Z) such that a − 1, d − 1, c ≡ 0 (mod N)\} .

We shall begin by recalling the geometric definition of modular forms over a field F that is given in [Ka2] and [Hi4].

If R is a ring in which N is invertible and E is an elliptic curve over R, we observe that a closed immersion t : Z/NZ ↪ E of group schemes over Spec R gives rise to a section s : Spec(R)⟶E of order N by restriction to the section 1 of Z/NZ.

Definition 1.1. An elliptic curve with Γ-level structure over a ring R is a pair (E, t) consisting of (1) an elliptic curve E over Spec(R);

(2) a closed immersion t : Z/NZ ↪ E of group schemes over Spec R. A triple (E, t, ω) where (E, t) is an elliptic curve with Γ-level structure and ω ∈ Ω_{E/R}^{1} is a global section of Ω_{E}^{1} over Spec(R) is called a marked elliptic curve with Γ-level structure.

The notion of R-isomorphisms between elliptic curves or marked elliptic curves with Γ-level structure is defined in the obvious way. Denote by Ell(Γ, R) the set of isomorphism classes of elliptic curves with Γ-level structure over R, and by Ẽll(Γ, R) the set of isomorphism classes of marked elliptic curves with Γ-level structure.

Definition 1.2. A weakly holomorphic algebraic modular form of weight k on Γ defined over a field F is a rule which to every isomorphism class of triples (E, t, ω) ∈ Ẽll(Γ, R) defined over an F -algebra R associates an element f(E, t, ω) ∈ R satisfying:

(1) (Compatibility with base change). For all F -algebra homomorphisms j : R⟶R′,

f((E, t, ω) ⊗_{j} R′) = j(f(E, t, ω)).

(2) (Weight k condition). For all λ ∈ R^{×},

f(E, t, λω) = λ^{−k}f(E, t, ω).

Let (Tate(q), t, ω_{can})_{/F ((q^{1/d}))} be the Tate elliptic curve G_{m}/q^{Z}, equipped with some level N structure t defined over F ((q^{1/d})) (for some d|N) and the canonical differential ω_{can} := \frac{du}{u} over F ((q)), where u is the usual parameter on G_{m}.

Definition 1.3. An algebraic modular form on Γ over F is a weakly holomorphic modular form satisfying

f (Tate(q), t, ω_{can}) belongs to F [[q^{1/d}]], for all t.

If these values belong to q^{1/d}F [[q^{1/d}]], then f is called a cusp form.

We denote by

S_{k}(Γ, F) ⊂ M_{k}(Γ, F) ⊂ M_{k}^{†}(Γ, F)

the F -vector spaces of cusp forms, algebraic modular forms, and weakly holomorphic modular forms respectively on Γ over F.

Write

C^{0} = Y_{1}(N), C = X_{1}(N) = Y_{1}(N) ∪ Z_{N},

for the usual modular curves over Q associated to Γ. The cuspidal subscheme Z_{N} is finite over Q. If N ≥ 3, the group Γ_{1}(N) is torsion-free and the curve C^{0} is a fine moduli scheme having a canonical smooth proper model over Spec(Z[1/N]). It represents the functor on Z[1/N]-algebras which to R associates the set Ell(Γ, R) of Definition 1.1. We will not make use of the integral model for now, and will view the curves C^{0} and C as defined over some base field F (of characteristic 0) for the rest of this chapter.

Let π : E⟶C^{0} be the universal elliptic curve with level N structure over C^{0}, and let \underline{ω} := π_{∗}Ω_{E/C^{0}}^{1} be

the line bundle of relative differentials on E/C^{0}. A weakly holomorphic modular form f ∈ M_{k}^{†}(Γ, F) can be viewed as a global section of the sheaf \underline{ω}^{k} over C^{0}, by setting

(1.1.1)  f(E, t) = f(E, t, ω)ω^{k},

where (E, t) is viewed as a point of C^{0}(R) and ω is an arbitrarily chosen generator (locally on Spec R) of Ω_{E/R}^{1}. Note that the expression on the right of (1.1.1) does not depend on the choice of ω.

Consider the relative de Rham cohomology sheaf on C^{0}:

L_{1} := ℝ^{1}π_{∗}(0 → O_{E} → Ω_{E/C^{0}}^{1} → 0).

It is a rank 2 algebraic vector bundle over C^{0} whose fibre at any geometric point x : Spec L⟶C^{0} is given by

(L_{1})_{x} := H_{dR}^{1}(E_{x}),

with E_{x} := E ×_{x} Spec L. There is a non-degenerate (Poincaré) pairing

⟨ , ⟩ : L_{1} × L_{1}⟶O_{C^{0}},

and the Hodge filtration on the fibres corresponds to an exact sequence of coherent sheaves over C^{0}:

(1.1.2)  0⟶\underline{ω}⟶L_{1}⟶\underline{ω}^{−1}⟶0.

The vector bundle L_{1} is also equipped with the canonical integrable Gauss-Manin connection

(1.1.3)  ∇ : L_{1}⟶L_{1} ⊗ Ω_{C^{0}}^{1}.

The Kodaira-Spencer map KS is defined to be the composite

KS : \underline{ω}⟶L_{1} ⟶^{∇} L_{1} ⊗ Ω_{C^{0}}^{1}⟶\underline{ω}^{−1} ⊗ Ω_{C^{0}}^{1},

in which the first and last arrows arise from (1.1.2). This map is an isomorphism of sheaves over C^{0}, and therefore gives rise to an identification

(1.1.4)  σ : \underline{ω}^{2} ⟶^{∼} Ω_{C^{0}}^{1},  σ(ω_{1} ⊗ ω_{2}) := ⟨ω_{1}, ∇ω_{2}⟩.

In addition to the geometric interpretation (1.1.1), it will also be convenient to view modular forms f ∈ M_{r+2}^{†}(Γ, F) as global sections of the sheaf \underline{ω}^{r} ⊗ Ω_{C^{0}}^{1}, by the rule

(1.1.5)  ω_{f}(E, t) := f(E, t, ω) · ω^{r} ⊗ σ(ω^{2}).

Assume for simplicity that all the cusps of X_{1}(N) are regular in the sense of [DS, §3.2]. (This condition is satisfied as soon as N > 4.) The line bundles \underline{ω} and L_{1} and their attendant structures extend naturally to the complete curve C as follows:

• The line bundle \underline{ω} admits an extension to C (denoted again by \underline{ω}) which is characterised by the

property

H^{0}(C, \underline{ω}^{k}) = M_{k}(Γ, F).

By Definition 1.3, the local sections of \underline{ω} in the neighborhood Spec F (ζ_{N})[[q^{1/d}]] of the cusp attached to the pair (Tate(q), q^{1/d}ζ_{N}) are expressions of the form hω_{can} with h ∈ F (ζ_{N})[[q^{1/d}]], where we recall that ω_{can} is the canonical differential on the Tate curve.

• The exact sequence (1.1.2), together with the given extensions of \underline{ω} and \underline{ω}^{−1} to C, determine an extension of L_{1} to C, in such a way that (1.1.2) becomes an exact sequence of sheaves over this base. The local sections of L_{1} in a neighborhood of the cusp (Tate(q), q^{1/d}ζ_{N}) are F (ζ_{N})[[q^{1/d}]]-linear combinations of ω_{can} and the local section ξ_{can} defined by

(1.1.6)  ∇ω_{can} =: ξ_{can} ⊗ \frac{dq}{q} .

(The sheaf L_{1} is described in Sec. 2.4. of [Schol1], where it is denoted E.)

• The Gauss-Manin connection ∇ of (1.1.3) extends to a connection with log poles

(1.1.7)  ∇ : L_{1}⟶L_{1} ⊗ Ω_{C}^{1}(log Z_{N}),

where Ω_{C}^{1}(log Z_{N}) denotes the sheaf of differentials on C with logarithmic singularities on the cuspidal subscheme Z_{N}. Over Spec F (ζ_{N})[[q^{1/d}]], it is described by the equation

(1.1.8)  ∇ω_{can} = ξ_{can} ⊗ \frac{dq}{q} ,  ∇ξ_{can} = 0.

• Finally, the Kodaira-Spencer isomorphism σ gives an identification

(1.1.9)  σ : \underline{ω}^{2} ⟶^{∼} Ω_{C}^{1}(log Z_{N})

of sheaves over C. Over Spec F (ζ_{N})[[q^{1/d}]], it is determined by

(1.1.10)  σ(ω_{can}^{2}) = \frac{dq}{q} .

• With these definitions, the rules (1.1.1) and (1.1.5) give identifications

(1.1.11)  M_{r+2}(Γ, F) = H^{0}(C, \underline{ω}^{r+2}) = H^{0}(C, \underline{ω}^{r} ⊗ Ω_{C}^{1}(log Z_{N})),
(1.1.12)  S_{r+2}(Γ, F) = H^{0}(C, \underline{ω}^{r} ⊗ Ω_{C}^{1}).

For any r ≥ 1, let

L_{r} := Sym^{r} L_{1}.

The sheaf L_{r} inherits from (1.1.2) a canonical Hodge filtration by sheaves of O_{C}-modules:

L_{r} ⊃ L_{r−1} ⊗ \underline{ω} ⊃ ⋯ ⊃ \underline{ω}^{r},

and the relative Poincaré duality

(1.1.13)  ⟨ , ⟩ : L_{r} × L_{r}⟶O_{C}

whose reduction to the geometric fibers is given by the rule

(1.1.14)  ⟨α_{1} ⋯ α_{r}, β_{1} ⋯ β_{r}⟩ = \frac{1}{r!} ∑_{σ∈S_{r}}⟨α_{1}, β_{σ1}⟩ ⋯ ⟨α_{r}, β_{σr}⟩,

where S_{r} denotes the symmetric group on r letters. The connection ∇ on L_{1} gives rise to a connection (which will also be denoted ∇)

∇ : L_{r}⟶L_{r} ⊗ Ω_{C}^{1}(log Z_{N}).

Let ˜∇ denote the composite

(1.1.15)  ˜∇ : L_{r} ⟶^{∇} L_{r} ⊗ Ω_{C}^{1}(log Z_{N}) ⟶^{id ⊗σ^{−1}} L_{r} ⊗ \underline{ω}^{2}⟶L_{r} ⊗ L_{2}⟶L_{r+2},

where the penultimate arrow is induced from (1.1.2) and the last arises from the natural projection

Sym^{r} ⊗ Sym^{2} ⟶ Sym^{r+2} .

The map ˜∇ (which, like ∇, is a homomorphism of abelian sheaves but not of O_{C}-modules) gives rise to differential operators on modular forms. More precisely, let

(1.1.16)  Ψ : L_{1}⟶\underline{ω}

be a splitting of the Hodge filtration (1.1.2), and let Ψ^{(k)} denote the corresponding homomorphism L_{k}⟶\underline{ω}^{k}. The splitting Ψ determines a differential operator

(1.1.17)  Θ_{Ψ} : M_{r}(Γ, F)⟶M_{r+2}(Γ, F),  (Θ_{Ψ}f)(E, t) := Ψ^{(r+2)}( ˜∇f)(E, t).

Example 1.4. We can construct a splitting Ψ as in (1.1.16) as follows. The datum of a pair (E, ω)_{/R} determines (locally on Spec R) canonical elements x ∈ H^{0}(E, O_{E}(2O_{E})) and y ∈ H^{0}(E, O_{E}(3O_{E})) satisfying

y^{2} = 4x^{3} + g_{2}x + g_{3}, for some g_{2}, g_{3} ∈ R, and \frac{dx}{y} = ω.

The decomposition

H_{dR}^{1}(E/R) = R [\frac{dx}{y}] ⊕ R [\frac{xdx}{y}]

determines a canonical algebraic (but not functorial) splitting Ψ_{alg} of the Hodge filtration on L_{1}. The resulting differential operator Θ_{alg} on M_{r}(Γ, F) is given in terms of q-expansions by the formula

Θ_{alg}(f) = θf − \frac{r}{12}P f, θ = q \frac{d}{dq} ,

where

P = 1 − 24 ∑_{n≥1} σ_{1}(n)q^{n}, (with σ_{1}(n) = ∑_{d|n} d)

arises from the Eisenstein series of weight 2. (Cf. §A1.4 of [Ka2].)

1.2. Modular forms over C. Assume now that F = C. The set C(C) of complex points of C is a compact Riemann surface, and the analytic map

pr : H⟶C^{0}(C), pr(τ) := (C/⟨1, τ⟩, \frac{1}{N} )

identifies C^{0}(C) with the quotient Γ\H, where we recall that Γ = Γ_{1}(N). The coherent sheaf L_{r} gives rise to an analytic sheaf L_{r}^{an} on the Riemann surface C(C); let L̃_{r}^{an} := pr^{∗} L_{r}^{an} denote its pullback to H.

Recall the elliptic fibration π : E⟶C^{0}, and let

𝕃_{1}^{B} := R^{1}π_{∗}Z, 𝕃_{r}^{B} := Sym^{r} 𝕃_{1}^{B},

be the locally constant sheaves of Z-modules whose fibers at x ∈ C^{0}(C) are identified with the Betti cohomology H_{B}^{1}(E_{x}, Z) and Sym^{r} H_{B}^{1}(E_{x}, Z) respectively. The local system

(1.2.1)  𝕃_{r} := 𝕃_{r}^{B} ⊗_{Z}C

is identified with the sheaf of horizontal sections of (L_{r}^{an}, ∇) over C^{0}(C). (Cf. [De1], thm. 2.17.)

A modular form f ∈ M_{k}^{†}(Γ, C) gives rise to a holomorphic section of \underline{ω}^{k} viewed as an analytic sheaf over C^{0}(C). It also gives rise to a holomorphic function on H by the rule

(1.2.2)  f(τ) := f (C/⟨1, τ⟩, \frac{1}{N} , 2πidw) ,

where w is the standard coordinate on C/⟨1, τ⟩. This function obeys the familiar transformation rule

(1.2.3)  f (\frac{aτ + b}{cτ + d}) = (cτ + d)^{k}f(τ),  for all ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{1}(N),

and the modular form f is completely determined by the associated function f(τ).

The Hodge filtration on H_{dR}^{1}(C/⟨1, τ⟩) admits a canonical, functorial (but non holomorphic) splitting

(1.2.4)  H_{dR}^{1}(C/⟨1, τ⟩) := Cdw ⊕ Cdw̄,

called the Hodge decomposition. In terms of the local coordinates τ, τ̄, dw, and dw̄, the Gauss-Manin connection and Kodaira-Spencer map are described by

(1.2.5)  ∇dw = (\frac{dw − dw̄}{τ − τ̄} ) ⊗ dτ,  σ((2πidw)^{2}) = 2πidτ.

The global sections of \underline{ω}^{r+2} and \underline{ω}^{r} ⊗ Ω_{C}^{1} attached to f in (1.1.1) and (1.1.5) are therefore given by the complex formulae

(1.2.6)  f (C/⟨1, τ⟩, \frac{1}{N} ) = f(τ)(2πidw)^{r+2}  ω_{f} (C/⟨1, τ⟩, \frac{1}{N} ) = f(τ)(2πidw)^{r} ⊗ (2πidτ).

Let L_{r}^{ra} denote the real analytic sheaf on C^{0} associated to L_{r}^{an} by forgetting the complex structure on C and retaining only its associated real analytic structure, and denote by \underline{ω}_{ra}^{r} the subsheaf of L_{r}^{ra} for the real analytic topology associated to \underline{ω}^{r}. The global sections of \underline{ω}_{ra}^{r} over C^{0} are called real analytic modular forms of weight r on Γ. They are identified, via (1.2.2), with real analytic functions on H satisfying the transformation property (1.2.3).

Following [Ka4], (1.8.3), we recall the Hodge decomposition of real analytic sheaves

(1.2.7)  L_{1}^{ra} = \underline{ω}_{ra} ⊕ ¯ \underline{ω}_{ra},

which induces (1.2.4) over the points of C^{0}(C). It gives rise to real analytic splittings

(1.2.8)  Ψ_{Hodge} : L_{1}^{ra}⟶\underline{ω}_{ra},  Ψ_{Hodge}^{(r)} : L_{r}^{ra}⟶\underline{ω}_{ra}^{r}.

A section f of \underline{ω}_{ra}^{r} which is of the form Ψ_{Hodge}^{(r)}(s) for some holomorphic section s of L_{r} over C is called a nearly holomorphic modular form on Γ. The holomorphic section s of L_{r} associated to a given nearly holomorphic modular form f is unique (cf. equation (5a) in §10.1 of [Hi2]). Following a common abuse of notation, a nearly holomorphic modular form is treated interchangeably as as a real analytic section f(τ)(2πidw)^{r} of \underline{ω}_{ra}^{r} and as a real analytic function f(τ) on H transforming under Γ like a modular form of weight r.

Let Θ_{Hodge} be the differential operator on nearly holomorphic modular forms associated to the splitting (1.2.8) as in (1.1.17), i.e., satisfying

Θ_{Hodge}(f) = Ψ_{Hodge}^{(r+2)}( ˜∇(s)), for all f = Ψ_{Hodge}^{(r)}(s) with s ∈ H^{0}(C, L_{r}).

The following lemma relates Θ_{Hodge} to the classical Shimura-Maass differential operator δ_{r} defined by

(1.2.9)  δ_{r}f(τ) := \frac{1}{2πi} (\frac{∂}{∂τ} + \frac{r}{τ − τ̄}) f(τ),

which maps real analytic modular forms of weight r to real analytic modular forms of weight r + 2.

Lemma 1.5. Let f be any nearly holomorphic modular form of weight r on Γ. Then

(1.2.10)  Θ_{Hodge}f = δ_{r}f.

Proof. Write f = Ψ_{Hodge}^{(r)}(s), where s is the holomorphic section of L_{r} giving rise to f, and expand s in terms of the local coordinates τ and w as

s = s_{0}(τ)dw̄^{r} + s_{1}(τ)dw̄^{r−1}dw + ⋯ + s_{r−1}(τ)dw̄dw^{r−1} + f(τ)(2πidw)^{r}.

Since s is a holomorphic section, its periods vary holomorphically, and therefore ∇s = ∇^{1,0}s, where ∇^{1,0} is the component of the Gauss-Manin connection on L_{r}^{ra} obtained by differentiating periods of real analytic sections in the holomorphic direction. Since the periods attached to the local section dw̄ are antiholomorphic, it follows that ∇^{1,0}(dw̄) = 0, and therefore, by (1.2.5), which continues to hold when ∇ is replaced by ∇^{1,0},

∇s = ∇^{1,0}s ≡ ∇^{1,0}(f(τ)(2πidw)^{r}) (mod dw̄H^{0}(C^{0}, L_{r−1} ⊗ Ω_{C}^{1}))
≡ (2πi)^{r} · (f_{τ} (τ)dw^{r} + rf(τ)dw^{r−1} (\frac{dw − dw̄}{τ − τ̄} )) ⊗ dτ,

where f_{τ} := \frac{∂f}{∂τ} is the derivative of f with respect to the holomorphic variable τ. It follows from the last identity in (1.2.5) and the definition of ˜∇ that

Ψ_{Hodge}^{(r+2)}( ˜∇(s)) = (2πi)^{r+1} · Ψ_{Hodge}^{(r+2)} (f_{τ} (τ)dw^{r+2} + rf(τ)dw^{r+1} (\frac{dw − dw̄}{τ − τ̄} ))
= δ_{r}f(τ)(2πidw)^{r+2}.

The lemma follows. □

More generally, letting

Θ_{Hodge}^{j} : f ↦ Ψ_{Hodge}^{(r+2j)}( ˜∇^{j}(s)),

one obtains Θ_{Hodge}^{j}(f) = δ_{r}^{j}f, where δ_{r}^{j} := δ_{r+2j−2} ∘ ⋯ ∘ δ_{r} is the j-th iterate of the Shimura-Maass derivative, sending nearly holomorphic modular forms of weight r to nearly holomorphic modular forms of weight r + 2j.

1.3. p-adic modular forms. A ring is called a p-adic ring if the natural homomorphism to its pro-p completion is an isomorphism. If R is a p-adic ring, then a triple (E, t, ω)_{/R} as in Definition 1.2 is said to be ordinary if the mod p reduction of E (viewed as an elliptic curve over R/pR) has invertible Hasse invariant. We briefly recall Katz’s definition of p-adic modular forms, which is modelled on Definition 1.2. In this definition we continue to assume that k is an integer ≥ 2.

Definition 1.6. A p-adic modular form of weight k on Γ defined over a p-adic ring Z is a rule which to every isomorphism class of ordinary triples (E, t, ω) ∈ Ẽll(Γ, R) defined over a p-adic Z-algebra R associates an element f(E, t, ω) ∈ R satisfying

(1) (Compatibility with base change). For all Z-algebra homomorphisms j : R⟶R′,

f((E, t, ω) ⊗_{j} R′) = j(f(E, t, ω)).

(2) (Weight k condition). For all λ ∈ R^{×},

f(E, t, λω) = λ^{−k}f(E, t, ω).

(3) (Behavior at the cusps). Let (Tate(q), t, ω_{can}) be the Tate elliptic curve G_{m}/q^{Z} equipped with any level N structure t defined over the p-adic completion of Z[ζ_{N}]((q^{1/d})), and the canonical differential ω_{can} over Z((q)). Then

f (Tate(q), t, ω_{can}) belongs to Z[ζ_{N}][[q^{1/d}]],

and f (Tate(q), t^{σ}, ω_{can}) = f (Tate(q), t, ω_{can})^{σ} for all σ ∈ Aut(Z(ζ_{N})/Z).

We will now recall the geometric interpretation of p-adic modular forms as sections of suitable rigid analytic line bundles. Assume that the prime p does not divide N, so that C extends to a canonical smooth proper model C over Spec Z_{p}. Write C_{F_{p}} := C ×_{Z_{p}} F_{p}, and let

red_{p} : C(C_{p})⟶C_{F_{p}}(F̄_{p})

denote the natural reduction map.

Let {P_{1}, …, P_{t}} be the finite subset of C_{F_{p}}(F̄_{p}) consisting of the supersingular points. The residue disc attached to P_{j}, denoted D(P_{j}), is the set of points of C(C_{p}) which have the same image as P_{j} under red_{p}. Let

C^{ord} = C(C_{p}) − D(P_{1}) − ⋯ − D(P_{t}).

Since the P_{j} are smooth points of C_{F_{p}}(F̄_{p}), the residue discs D(P_{j}) are conformal to the open unit disc U ⊂ C_{p} consisting of z ∈ C_{p} with |z| < 1. The set C^{ord} is an example of an affinoid subset of C(C_{p}) with good reduction. (These concepts are discussed in somewhat more detail in Section 3.5. For general definitions and a more systematic discussion, see also, for example, Sections II and III of [Col2].)

The algebraic vector bundle L_{r} on C gives rise to a rigid analytic coherent sheaf L_{r}^{rig} on C^{ord}, equipped with the Gauss-Manin connection

∇ : L_{r}^{rig}⟶L_{r}^{rig} ⊗ Ω^{1}(log Z_{N}),

and a subsheaf \underline{ω}^{r} for the rigid analytic topology on C^{ord}. A p-adic modular form f of weight r for Γ corresponds, via (1.1.1), to a rigid analytic section of \underline{ω}^{r} over C^{ord}.

Following [Ka4], Theorem (1.11.27), there is a unique decomposition of rigid analytic sheaves

(1.3.1)  L_{1}^{rig} = \underline{ω} ⊕ L_{1}^{Frob}

such that the Frobenius endomorphism preserves (and acts invertibly) on L_{1}^{Frob}. In the p-adic theory, this “unit root” decomposition plays a role analogous to that of the Hodge decomposition in the complex setting. Most importantly, (1.3.1) gives rise to a rigid analytic splitting over C^{ord}

Ψ_{Frob} : L_{1}^{rig}⟶\underline{ω}.

Let Θ_{Frob} be the differential operator associated to this splitting as in (1.1.17). It maps p-adic modular forms of weight r to p-adic modular forms of weight r + 2. The following lemma relates Θ_{Frob} to the classical Atkin-Serre theta operator whose effect on q-expansions f(Tate(q), ζ_{N}, ω_{can}) = ∑ a_{n}q^{n} is given by

(1.3.2)  θf(Tate(q), ζ_{N}, ω_{can}) = q \frac{d}{dq} ∑_{n=1}^{∞} a_{n}q^{n} = ∑_{n=1}^{∞} na_{n}q^{n}.

Lemma 1.7. For all p-adic modular forms f of weight r,

(1.3.3)  Θ_{Frob}f = θf.

Proof. Since a p-adic modular form is determined by its q-expansion, it is enough to check the identity on the Tate curve. By (1.1.8),

∇f (Tate(q), ζ_{N}) = ∇(f(q)ω_{can}^{r})
= (q \frac{d}{dq} f(q)ω_{can}^{r} + rf(q)ω_{can}^{r−1}ξ_{can}) \frac{dq}{q} .

Therefore, by (1.1.10),

(1.3.4)  ˜∇f (Tate(q), ζ_{N}) = q \frac{d}{dq} f(q)ω_{can}^{r+2} + rf(q)ω_{can}^{r+1}ξ_{can}.

Since the Frobenius endomorphism respects the Gauss-Manin connection, it preserves the line spanned by the unique horizontal section ξ_{can} of L_{1} over Z′[[q]], and therefore ξ_{can} is stable under Frobenius. (Cf. Sec. A2.2 of [Ka2].) It follows that the unit root subspace of the Tate curve Tate(q) over the p-adic completion R of Z′((q)) is equal to

H_{dR}^{1}(Tate(q))^{Frob} = Rξ_{can}.

Hence Ψ_{Frob}(ξ_{can}) = 0. Applying Ψ_{Frob}^{(r+2)} to equation (1.3.4) shows that

Θ_{Frob}f(Tate(q), ζ_{N}, ω_{can}) = θf(Tate(q), ζ_{N}, ω_{can}).
□

1.4. Elliptic curves with complex multiplication. Let K be an imaginary quadratic field of discriminant −d_{K}, let O_{K} be its ring of integers, and let H denote the Hilbert class field of K. Let A be a fixed elliptic curve defined over H satisfying

End_{H}(A) ≃ O_{K}.

The identification O_{K} = End_{H}(A) is normalised so that the endomorphism [α] induces multiplication by α on Ω_{A/H}^{1}.

Cohomology. The Hodge filtration on the de Rham cohomology H_{dR}^{1}(A/F) (over any field F which contains H) admits a canonical, functorial algebraic splitting

(1.4.1)  H_{dR}^{1}(A/F) = H_{dR}^{1,0}(A/F) ⊕ H_{dR}^{0,1}(A/F),

which agrees with the Hodge decomposition of H_{dR}^{1}(A/C) when F = C and with the unit root decomposition over a p-adic ring when A is ordinary. This decomposition is characterised by the conditions

H_{dR}^{1,0}(A/F) = Ω_{A/F}^{1} , λ^{∗}η = λ^{ρ}η, ∀ λ ∈ O_{K}, η ∈ H_{dR}^{0,1}(A/F),

where λ ↦ λ^{ρ} is the non-trivial automorphism of K. The choice of a non-zero differential ω_{A} ∈ Ω_{A/F}^{1} = H_{dR}^{1,0}(A/F) thus determines a generator η_{A} of H_{dR}^{0,1}(A/F) satisfying

(1.4.2)  ⟨ω_{A}, η_{A}⟩ = 1,

where ⟨ , ⟩ denotes the algebraic cup product pairing on de Rham cohomology.

Let S_{r} denote the symmetric group on r letters. Multiplication by −1 on A, combined with the natural permutation action of S_{r} on A^{r}, gives rise to an action of the wreath product

(1.4.3)  Ξ_{r} := (μ_{2})^{r} ⋊ S_{r}

on A^{r}. Let j : Ξ_{r}⟶μ_{2} be the homomorphism which is the identity on μ_{2} and the sign character on S_{r}, and let

(1.4.4)  ϵ_{A} := \frac{1}{2^{r}r!} ∑_{σ∈Ξ_{r}} j(σ)σ ∈ Q[Aut(A^{r})]

denote the associated idempotent in the rational group ring of Aut(A^{r}). By functoriality, it induces an idempotent on H_{dR}^{∗}(A^{r}/F). Recall the Künneth decomposition

(1.4.5)  H_{dR}^{∗}(A^{r}/F) = ⊕_{(i_{1},...,i_{r})}H_{dR}^{i_{1}}(A/F) ⊗ ⋯ ⊗ H_{dR}^{i_{r}}(A/F),

where the direct sum is taken over all r-tuples (i_{1}, …, i_{r}) with 0 ≤ i_{j} ≤ 2. The natural action of S_{r} on H_{dR}^{1}(A/F)^{⊗r} gives rise to a subspace Sym^{r} H_{dR}^{1}(A/F) consisting of classes which are fixed by this action.

Lemma 1.8. The image of the projector ϵ_{A} acting on H_{dR}^{∗}(A^{r}/F) is equal to Sym^{r} H_{dR}^{1}(A/F). More precisely,

ϵ_{A}H_{dR}^{j}(A^{r}/F) = \{ \begin{matrix}0 & if j ≠ r; \\ Sym^{r} H_{dR}^{1}(A/F) & if j = r. \end{matrix}

Proof. Since multiplication by (−1) acts as −1 on H_{dR}^{1}(A/F) and as 1 on H_{dR}^{0}(A/F) and H_{dR}^{2}(A/F), it follows that ϵ_{A} annihilates all the terms in the Künneth decomposition (1.4.5) except H_{dR}^{1}(A/F)^{⊗r}. The natural action of S_{r} on this term corresponds to the geometric permutation action of S_{r} on A^{r}, twisted by the sign character. It follows that the restriction of ϵ_{A} to H_{dR}^{1}(A/F)^{⊗r} induces the natural projection onto the space Sym^{r} H_{dR}^{1}(A/F) of symmetric tensors. □

For any j such that 0 ≤ j ≤ r, we define ω_{A}^{j}η_{A}^{r−j} by

(1.4.6)  ω_{A}^{j}η_{A}^{r−j} := ϵ_{A}^{∗}(p_{1}^{∗}ω_{A} ∧ ⋯ p_{j}^{∗}ω_{A} ∧ p_{j+1}^{∗}η_{A} ∧ ⋯ ∧ p_{r}^{∗}η_{A})
= \frac{j!(r − j)!}{r!} ∑_{I⊂\{1,...,r\}} p_{1}^{∗}ϖ_{1,I} ∧ … ∧ p_{r}^{∗}ϖ_{r,I},

where ϖ_{i,I} := ω_{A} or η_{A} according as i ∈ I or i ∉ I.

Note that the classes ω_{A}^{j}η_{A}^{r−j} form a basis of the vector space

ϵ_{A}H_{dR}^{r}(A^{r}/F) = Sym^{r} H_{dR}^{1}(A/F).

Isogenies. It will always be assumed that A satisfies the following “Heegner hypothesis” relative to a fixed positive integer N that is mentioned in (0.0.3) of the Introduction.

Assumption 1.9. There is an ideal 𝔑 of O_{K} of norm N such that O_{K}/𝔑 = Z/NZ. (Such an ideal is called a cyclic ideal of norm N in O_{K}.)

Since both A and its endomorphisms are defined over the Hilbert class field H, the group scheme A[𝔑] of 𝔑-torsion in A is a cyclic subgroup scheme of A of order N defined over this field. The absolute Galois group G_{H} acts naturally on its set of geometric points. Let H̃ be the smallest extension of H over which this Galois representation becomes trivial. The choice of a section t_{A} : Spec(H̃)⟶A[𝔑] of order N gives rise to a Γ-level structure on A defined over any field F that contains H̃. Fix such a t_{A} once and for all.

Consider the set of pairs (φ, A′), where A′ is an elliptic curve and φ : A⟶A′ is an isogeny (defined over K̄). Two pairs (φ_{1}, A′_{1}) and (φ_{2}, A′_{2}) are said to be isomorphic if there is a K̄-isomorphism ι : A′_{1}⟶A′_{2} satisfying ιφ_{1} = φ_{2}. Let

Isog(A) := {Isomorphism class of pairs (φ, A′)} .

The absolute Galois group G_{K} = Gal(K̄/K) acts naturally on Isog(A) and a pair (φ, A′) admits a representative defined over a field F ⊂ K̄ if it is fixed by the group G_{F} ⊂ G_{K}. Fix (φ, A′) ∈ Isog(A). Since A has complex multiplication by O_{K}, the endomorphism ring of A′ is an order in O_{K}. Such an order is completely determined by its conductor, and therefore there is a unique integer c ≥ 1 such that End_{F} (A′) = O_{c} := Z + cO_{K}. A pair (φ, A′) is said to be of conductor c if End_{F} (A′) = O_{c}. Clearly this notion is well defined on isomorphism classes, and hence we may set

Isog_{c}(A) := {Isomorphism classes of pairs (φ, A′) of conductor c} .

More generally, let Isog^{𝔑}(A) be the subset of Isog(A) consisting of pairs (φ, A′) where φ is an isogeny whose kernel intersects A[𝔑] trivially, and set Isog_{c}^{𝔑}(A) := Isog_{c}(A) ∩ Isog^{𝔑}(A).

Let P_{K}(O_{c}) denote the group of projective rank one O_{c}-submodules of K and let P (O_{c}) denote the subsemigroup of modules that are contained in O_{c} and are relatively prime to 𝔑_{c} := 𝔑 ∩ O_{c}. The semigroup P (O_{c}) acts naturally on Isog_{c}(A) and Isog_{c}^{𝔑}(A) by the rule 𝔞 ∗ (φ, A′) = (φ_{𝔞}φ, A′_{𝔞}), where

(1.4.7)  φ_{𝔞} : A′⟶A′_{𝔞} := A′/A′[𝔞]

is the natural isogeny. Note that, if 𝔞 = O_{c} · a is free, then 𝔞 ∗ (φ, A′) = (aφ, A′).

Let (A_{1}, t_{1}, ω_{1}) and (A_{2}, t_{2}, ω_{2}) be two marked elliptic curves with Γ-level structure. The following notion of an isogeny

φ : (A_{1}, t_{1}, ω_{1})⟶(A_{2}, t_{2}, ω_{2})

will be convenient from the notational viewpoint.

Definition 1.10. An isogeny from (A_{1}, t_{1}, ω_{1}) to (A_{2}, t_{2}, ω_{2}) is an isogeny φ : A_{1}⟶A_{2} on the underlying elliptic curves satisfying

φ(t_{1}) = t_{2}, φ^{∗}(ω_{2}) = ω_{1}.

The action of P (O_{c}) on Isog_{c}^{𝔑}(A) that was just defined gives rise to an action of P (O_{c}) on the set of isomorphism classes of triples (A′, t′, ω′) with End(A′) = O_{c} and t′ ∈ A′[𝔑_{c}], by the rule

(1.4.8)  𝔞 ∗ (A′, t′, ω′) = (A′_{𝔞}, φ_{𝔞}(t′), ω′_{𝔞}),  where φ_{𝔞}^{∗}(ω′_{𝔞}) = ω′.

Remark 1.11. Let 𝔸_{K,f} denote the ring of finite adèles of K and let Ô_{c} denote (O_{c} ⊗ Ẑ), viewed as a subring of 𝔸_{K,f}. The group P_{K}(O_{c}) is naturally identified with 𝔸_{K,f}^{×}/Ô_{c}^{×}, by associating to 𝔞 a generator (a_{v}) ∈ 𝔸_{K,f}^{×} of 𝔞 ⊗_{O_{c}} Ô_{c}.

1.5. Values of modular forms at CM points. Following the notations of Section 1.4, we continue to let (A, t_{A}, ω_{A}) be a marked elliptic curve with Γ-level structure and complex multiplication by O_{K}, defined over a field F, and let φ : (A, t_{A}, ω_{A})⟶(A′, t′, ω′) be an isogeny of marked elliptic curves over F.

Fix complex and p-adic embeddings ι_{∞} : F ⟶C and ι_{p} : F ⟶C_{p}, and use these to view A and A′ as curves over C and O_{C_{p}} (by fixing a good integral model) respectively. If f belongs to the space M_{k}^{†}(Γ, F) of modular forms over F, then by definition f(A′, t′, ω′) belongs to F as well. Note that f can be viewed as a p-adic modular form, after possibly rescaling it. The following algebraicity theorem asserts that a similar conclusion holds for Θ_{Hodge}(f) and Θ_{Frob}(f), evaluated on ι_{∞}(A′, t′, ω′) and ι_{p}(A′, t′, ω′) respectively.

Proposition 1.12. Let (A′, t′, ω′)_{/F} be a marked elliptic curve with complex multiplication by an order in K. Assume that A′, viewed as an elliptic curve over O_{C_{p}}, is ordinary. Then:

(1) The complex number Θ_{Hodge}f(A′, t′, ω′) belongs to ι_{∞}(F).

(2) The p-adic number Θ_{Frob}f(A′, t′, ω′) belongs to ι_{p}(F).

(3) Viewing these two quantities as elements of F, we have

Θ_{Hodge}f(A′, t′, ω′) = Θ_{Frob}f(A′, t′, ω′).

Proof. Part (1) is due to Shimura [Shim1] and parts (2) and (3) are due to Katz [Ka4]. Our proof below follows Katz’s approach. (See also the article of Hida [Hi4].) The key point is that any endomorphism α ∈ O_{K} of A′ respects the algebraic splitting of the Hodge filtration on H_{dR}^{1}(A′/F) defined in equation (1.4.1) of Section 1.4, and acts on H_{dR}^{0,1}(A′/F) via multiplication by ᾱ. It follows that H_{dR}^{1}(A′/F) = Ω^{1}(A′/F) ⊕ H_{dR}^{0,1}(A′/F) agrees with the Hodge decomposition of H_{dR}^{1}(A′ ⊗_{ι_{∞}} C) and with the unit root decomposition of H_{dR}^{1}(A′ ⊗_{ι_{p}} C_{p}), which both share this property. More precisely,

H_{dR}^{0,1}(A′/F) ⊗_{ι_{∞}} C = H_{dR}^{0,1}(A′ ⊗_{ι_{∞}} C),
H_{dR}^{0,1}(A′/F) ⊗_{ι_{p}} C_{p} = H_{dR}^{1}(A′ ⊗_{ι_{p}} C_{p})^{Frob}.

Therefore, Ψ_{Hodge}^{(r+2)} ˜∇f(A′, t′) and Ψ_{Frob}^{(r+2)} ˜∇f(A′, t′) both belong to Sym^{r+2} Ω^{1}(A′/F), and are equal. The proposition follows. □

2. Generalised Heegner cycles

2.1. Kuga-Sato varieties. Let π : E⟶C be the universal generalised elliptic curve with Γ_{1}(N) level structure, extending the universal elliptic curve over C^{0} introduced in Section 1.1, which exists because our running assumption that N > 4. The variety W_{1} := E is smooth and proper, and the geometric fibers of π above a closed point x ∈ C are singular precisely when x is a cusp. The geometric fiber π^{−1}(x) is then isomorphic to a chain of projective lines intersecting at ordinary double points whose dual graph is an m-gon for a suitable m|N, depending on x. Let W_{1}^{∗} ⊂ W_{1} denote the relative identity component of the Néron model of E over X_{1}(N), whose geometric fibers above the cusps are isomorphic to the multiplicative group G_{m}.

Fix an integer r ≥ 0, and let

W_{r}^{∗} := W_{1}^{∗} ×_{C} W_{1}^{∗} ×_{C} ⋯ ×_{C} W_{1}^{∗} ⊂ W_{r}^{♯} := E ×_{C} E ×_{C} ⋯ ×_{C} E,

denote the r-fold fiber products of W_{1}^{∗} and E respectively over C.

Write W_{r} for the canonical desingularisation of W_{r}^{♯}, as described in [De2], Lemmas 5.4 and 5.5, and [Schol2] 1.0.3, for example. In these references, these constructions are performed for the universal elliptic curve over the modular curve X(N) with full level N structure, but can be adapted to deal with the case of X_{1}(N); see the Appendix for further details on this more general construction, even over Spec Z[\frac{1}{N}].

Denote by

W_{r}^{0} := W_{r} ×_{C} C^{0} = W_{r}^{♯} ×_{C} C^{0} = W_{r}^{∗} ×_{C} C^{0}

the complement in W_{r} of the geometric fibers above the cusps, and let W_{r}^{reg} ∈ W_{r}^{♯} be the locus where the natural projection W_{r}^{♯}⟶C is smooth. As in 1.3.2. of [Schol2] there is a non-canonical isomorphism

(2.1.1)  W_{r}^{reg} ×_{C} Z_{∞} = ∐_{d|N} (Z_{∞}(d) × (G_{m} × Z/dZ)^{r}) ,

where Z_{∞} ⊂ C denotes the cuspidal subscheme and Z_{∞}(d) ⊂ Z_{∞} is the (possibly empty) subscheme of cusps with ramification degree d over the modular curve of level one. The varieties E, C, W_{r}^{♯}, W_{r}^{∗}, W_{r} and W_{r}^{0} are all defined over Q, and can therefore be viewed as defined over any field F of characteristic 0. It will be convenient to fix such an F at the outset.

Translation by the sections of order N gives rise to an action of (Z/NZ)^{r} on W_{r}^{♯}, which extends to W_{r} by the canonical nature of the desingularisation. The group (Z/NZ)^{r} also acts transitively (but not freely, in general) on the set of components of W_{r}^{♯} above any cusp of C arising in (2.1.1). Let σ_{a} denote the automorphism of W_{r} associated to a ∈ (Z/NZ)^{r}, and let

ϵ_{W}^{(1)} = \frac{1}{N^{r}} ∑_{a∈(Z/NZ)^{r}} σ_{a}

denote the corresponding idempotent in the rational group ring of (Z/NZ)^{r}. Similarly, the group Ξ_{r} of (1.4.3) can be viewed as a subgroup of Aut(W_{r}/C) acting on the fibers of the natural projection from W_{r} to C. Let ϵ_{W}^{(2)} be the idempotent in the group ring Z[1/2r!][Aut(W_{r}/C)] which is defined by the same formula as in (1.4.4) with A^{r} replaced by W_{r}/C. The idempotents ϵ_{W}^{(1)} and ϵ_{W}^{(2)} commute, and therefore the composition

(2.1.2)  ϵ_{W} = ϵ_{W}^{(2)}ϵ_{W}^{(1)}

defines a projector in the ring of rational correspondences on W_{r}.

Let

Ω^{0}(L_{r}) := L_{r}, Ω^{1}(L_{r}) := L_{r} ⊗ Ω_{C}^{1} + ∇(L_{r}).

The complex

(2.1.3)  0⟶Ω^{0}(L_{r}) ⟶^{∇} Ω^{1}(L_{r})⟶0

of sheaves over C is the smallest subcomplex of

(2.1.4)  0⟶L_{r} ⟶^{∇} L_{r} ⊗ Ω_{C}^{1}(log Z_{N})⟶0

which contains L_{r} and L_{r} ⊗ Ω_{C}^{1} in degrees 0 and 1 respectively. The de Rham cohomology of C attached to L_{r}, denoted H_{dR}^{i}(C/F, L_{r}, ∇), is defined to be the ith hypercohomology of the complex (2.1.4):

H_{dR}^{i}(C/F, L_{r}, ∇) := ℍ^{i}(C/F, L_{r} ⊗ Ω^{•}(log Z_{N})).

The parabolic de Rham cohomology of C attached to L_{r} is defined, following Section 2.6 of [Schol1] as the hypercohomology of the subcomplex (2.1.3):

H_{par}^{i}(C/F, L_{r}, ∇) := ℍ^{i}(C/F, Ω^{•}(L_{r})).

In degree 0, we have

H_{par}^{0}(C/F, L_{r}, ∇) = H_{dR}^{0}(C/F, L_{r}, ∇).

As explained in the proof of Theorem 2.7 (i) of [Schol1], the parabolic cohomology H_{par}^{1}(C/F, L_{r}, ∇) in degree 1 is equipped with a natural filtration

(2.1.5)  0⟶H^{0}(C/F, \underline{ω}^{r} ⊗ Ω_{C}^{1})⟶H_{par}^{1}(C/F, L_{r}, ∇)⟶H^{1}(C/F, \underline{ω}^{−r})⟶0.

The de Rham cohomology groups H_{dR}^{i}(X/F) (attached to any variety X over F) and H_{dR}^{i}(C/F, L_{r}, ∇) will sometimes be abbreviated to H_{dR}^{i}(X) and H_{dR}^{i}(C, L_{r}, ∇), and likewise for the parabolic cohomology groups, when no confusion results from suppressing the field of definition F in the notation.

Lemma 2.1. If r = 0, then H_{dR}^{0}(C, L_{r}, ∇) = F, and H_{dR}^{0}(C, L_{r}, ∇) = 0 otherwise.

Proof. Fix an embedding of F into C and consider H_{dR}^{0}(C/C, L_{r}, ∇) = H_{dR}^{0}(C/F, L_{r}, ∇) ⊗_{F} C. By the GAGA principle,

H_{dR}^{0}(C/C, L_{r}, ∇) = H_{dR}^{0}(C, L_{r}^{an}, ∇).

The restriction map

H_{dR}^{0}(C, L_{r}^{an}, ∇)⟶H_{dR}^{0}(C^{0}, L_{r}^{an}, ∇)

is injective, and

H_{dR}^{0}(C^{0}, L_{r}^{an}, ∇) = H^{0}(C^{0}, 𝕃_{r}),

where 𝕃_{r} is the local system introduced in (1.2.1). This local system corresponds to the r-th symmetric power of the standard two-dimensional representation C^{2} of Γ ⊂ SL_{2}(Z) ⊂ SL_{2}(C), and therefore

H^{0}(C^{0}, 𝕃_{r}) = H^{0}(Γ, Sym^{r}(C^{2})) = \{ \begin{matrix}C & if r = 0, \\ 0 & otherwise. \end{matrix}

The lemma follows. □

We wish to describe the image of ϵ_{W} on the (middle) cohomology of W_{r}, and relate this image to H_{par}^{1}(C, L_{r}, ∇).

Lemma 2.2. Assume r ≥ 1.

(1) The image of ϵ_{W}^{(2)} (and of ϵ_{W}) acting on H_{dR}^{∗}(W_{r}^{0}/F) is canonically isomorphic to H_{dR}^{1}(C, L_{r}, ∇).

(2) The image of ϵ_{W} acting on H_{dR}^{∗}(W_{r}/F) is canonically isomorphic to H_{par}^{1}(C, L_{r}, ∇).

(3) Furthermore, the Hodge filtration on ϵ_{W} H_{dR}^{∗}(W_{r}/F) = ϵ_{W} H_{dR}^{r+1}(W_{r}/F) is given by (2.1.5), i.e.,

Fil^{0} = H_{par}^{1}(C, L_{r}, ∇),
Fil^{1} = Fil^{2} = ⋯ = Fil^{r+1} = H^{0}(C, \underline{ω}^{r} ⊗ Ω_{C}^{1}),
Fil^{r+2} = ⋯ = 0,
where Fil^{j} denotes the j-th step in the Hodge filtration on ϵ_{W} H_{dR}^{r+1}(W_{r}).

Proof. The arguments below are mild adaptations of those in [Schol2], [Schol3].

(1) By [De1] Cor. 3.15, the natural map

H_{dR}^{i}(C, L_{r}, ∇) → H_{dR}^{i}(C^{0}, L_{r}|_{C^{0}}, ∇) := ℍ^{i}(C^{0}, Ω^{.}(L_{r})|_{C^{0}})

is an isomorphism. Consider the Leray spectral sequence for de Rham cohomology ([Ka1], Remark 3.3) applied to the map W_{r}^{0} → C^{0}: i.e.,

E_{2}^{pq} = H_{dR}^{p}(C^{0}, H_{dR}^{q}(W_{r}^{0}/C^{0}), ∇) ⇒ H_{dR}^{p+q}(W_{r}^{0}).

By the same argument as in [De2] Lemma 5.3, this spectral sequence degenerates at E_{2} and identifies the space H_{dR}^{p}(C^{0}, H_{dR}^{q}(W_{r}^{0}/C^{0}), ∇) with the subspace of H_{dR}^{p+q}(W_{r}^{0}) on which [m] acts as m^{q}. (Here [m] denotes multiplication by m on the fibers of W_{r}^{0}/C^{0}.) Applying the projector ϵ_{W}^{(2)}, we find

ϵ_{W}^{(2)}H_{dR}^{∗}(W_{r}^{0}/C^{0}) = ϵ_{W}^{(2)}H_{dR}^{r}(W_{r}^{0}/C^{0}) = L_{r}|_{C^{0}}

and

H_{dR}^{1}(C^{0}, L_{r}|_{C^{0}}, ∇) ≃ ϵ_{W}^{(2)}H_{dR}^{r+1}(W_{r}^{0}) = ϵ_{W}^{(2)}H_{dR}^{∗}(W_{r}^{0}).

A similar statement holds with ϵ_{W}^{(2)} replaced by ϵ_{W}, since translation by W_{r}^{0}(C^{0}) on H_{dR}^{1}(W_{r}^{0}/C^{0}) is trivial (as W_{r}^{0} → C^{0} is an abelian scheme).

(2) We use the following fact due to Scholl: there is a canonical isomorphism

ϵ_{W} H_{.}^{i}(W_{r}) ≃ ϵ_{W}^{(2)}H_{.}^{i}(W_{r}^{∗}),

for · = B, et or dR. This is proved in [Schol2], Thm. 3.1.0 for the case of full level structure, and the modifications needed to extend this to X_{1}(N) are described in [Schol3], §2, especially §2.9-2.12. Now consider the Gysin sequence for the inclusion W_{r}^{0} ↪ W^{∗}, writing Z := W_{r}^{∗} \ W_{r}^{0}:

→ H^{i}(W^{∗}) → H^{i}(W_{r}^{0}) → H^{i−1}(Z)(−1) → H^{i+1}(W^{∗}) →

Since (by [Schol2], Lemma 1.3.1 (i) and [Schol3] §2.9)

ϵ_{W}^{(2)}H^{i}(Z) = \{ \begin{matrix}0, if i ≠ r; \\ H^{0}(Z_{∞})(−r), if i = r, \end{matrix}

we see from part (1) of the lemma that ϵ_{W}^{(2)}H^{i}(W^{∗}) = 0 for i ≠ r + 1, r + 2. Further there is an exact sequence (in any cohomology theory)

[figure: 0; ϵ_{W}^{(2)}H^{r+1}(W_{r}^{∗}); ϵ_{W}^{(2)}H^{r+1}(W_{r}^{0}); ρ; H^{0}(Z_{∞})(−r − 1); σ; ϵ_{W}^{(2)}H^{r+2}(W_{r}^{∗}); 0; ϵ_{W} H^{r+1}(W_{r}); ϵ_{W} H^{r+2}(W_{r})]

The map σ vanishes since its source and image are pure of weight 2r + 2 and r + 2 respectively, and r ≠ 0, hence ϵ_{W} H^{r+2}(W_{r}) = 0. In the de Rham realization, we have from part (1) that ϵ_{W}^{(2)}H^{r+1}(W_{r}^{0}) = H_{dR}^{1}(C, L_{r}, ∇) = H_{dR}^{1}(C^{0}, L_{r}|_{C^{0}}, ∇) and hence ϵ_{W}^{(2)}H_{dR}^{r+1}(W_{r}^{∗}) is identified naturally with the kernel of the map

H_{dR}^{1}(C^{0}, L_{r}|_{C^{0}}, ∇) ⟶^{ρ_{dR}} H_{dR}^{0}(Z_{∞}, −r − 1),

which is just H_{par}^{1}(C, L_{r}, ∇).

(3) See Thm. 2.7 (i) and Remark 2.8 of [Schol1] . □

Corollary 2.3. The assignment

f ↦ ω_{f} = f(E, t, ω)ω^{r} ⊗ σ(ω^{2})

induces an identification

S_{r+2}(Γ, F) ⟶^{∼} Fil^{r+1} ϵ_{W} H_{dR}^{r+1}(W_{r}/F).

Proof. This follows from Part 2 of Lemma 2.2 combined with (1.1.12) (the case r = 0 being well known). □

2.2. The variety X_{r} and its cohomology. Recall that A is the elliptic curve with complex multiplication by O_{K} that was fixed in Section 1.4, defined over the Hilbert class field H of K. Fix a field F ⊃ H, and, for each r ≥ 0, consider the (2r+1)-dimensional variety over F given by

X_{r} := W_{r} × A^{r}.

Like the Kuga-Sato variety W_{2r}, the variety X_{r} is equipped with a proper morphism

π_{r} : X_{r}⟶C

with 2r-dimensional fibers. The fibers above points of C^{0} are products of elliptic curves of the form E^{r} × A^{r}, where E varies and A is fixed.

The projectors ϵ_{A} and ϵ_{W} defined in (1.4.4) and (2.1.2) respectively give rise to commuting idempotents in the ring of correspondences on X_{r} which preserve the fibers of the projection π_{r} : X_{r}⟶C. We set

(2.2.1)  ϵ_{X} := ϵ_{W} ϵ_{A}.

By functoriality, the idempotent ϵ_{X} acts as a projector on the various cohomology groups associated to the variety X_{r}.

We define a coherent sheaf of O_{C}-modules by setting

(2.2.2)  L_{r,r} = L_{r} ⊗ Sym^{r} H_{dR}^{1}(A).

Note that L_{r,r} is equipped with the self-duality

(2.2.3)  ⟨ , ⟩ : L_{r,r} × L_{r,r}⟶O_{C}

arising from Poincaré duality on the fibers. It is described explicitly in terms of equation (1.1.14) and its analogue for Sym^{r} H_{dR}^{1}(A). Let

(2.2.4)  𝕃_{r,r} := 𝕃_{r} ⊗ Sym^{r} H_{dR}^{1}(A/C)

denote the corresponding locally constant sheaf (for the complex topology on C^{0}(C)). The sheaf 𝕃_{r,r} is the sheaf of horizontal sections of L_{r,r}^{an} relative to the Gauss-Manin connection

∇ : L_{r,r}⟶L_{r,r} ⊗ Ω_{C}^{1}(log Z_{N}).

This connection is induced by the Gauss-Manin connection on L_{r} combined with the trivial connection on H_{dR}^{1}(A). The de Rham cohomology attached to (L_{r,r}, ∇) is defined in the same way as for (L_{r}, ∇), and one has

H_{dR}^{1}(C, L_{r,r}, ∇) = H_{dR}^{1}(C, L_{r}, ∇) ⊗ Sym^{r} H_{dR}^{1}(A),
H_{par}^{1}(C, L_{r,r}, ∇) = H_{par}^{1}(C, L_{r}, ∇) ⊗ Sym^{r} H_{dR}^{1}(A).

Proposition 2.4. Assume that r is ≥ 1. The image of the projector ϵ_{X} acting on H_{dR}^{∗}(X_{r}) is given by

ϵ_{X}H_{dR}^{∗}(X_{r}) = H_{par}^{1}(C, L_{r,r}, ∇) = H_{par}^{1}(C, L_{r}, ∇) ⊗ Sym^{r} H_{dR}^{1}(A).

In particular,

ϵ_{X}H_{dR}^{j}(X_{r}) = \{ \begin{matrix}0 & if j ≠ 2r + 1; \\ H_{par}^{1}(C, L_{r,r}, ∇) & if j = 2r + 1. \end{matrix}

Furthermore, if Fil^{j} denotes the j-th step in the Hodge filtration on ϵ_{X}H_{dR}^{2r+1}(X_{r}), then

(2.2.5)  Fil^{r+1} = H^{0}(C, \underline{ω}^{r} ⊗ Ω_{C}^{1}) ⊗ Sym^{r} H_{dR}^{1}(A).

Proof. This follows directly from Lemmas 1.8 and 2.2 in light of the Künneth decomposition for the cohomology of X_{r} = W_{r} × A^{r}. □

Proposition 2.5. The assignment f ⊗ α ↦ ω_{f} ∧ α induces an identification

S_{r+2}(Γ, F) ⊗ Sym^{r} H_{dR}^{1}(A/F) = Fil^{r+1} ϵ_{X}H_{dR}^{2r+1}(X_{r}/F).

Proof. This follows directly from Corollary 2.3, combined with Proposition 2.4 when r ≥ 1. □

Given any integer 0 ≤ j ≤ r, note in particular that the class

ω_{f} ∧ ω_{A}^{j}η_{A}^{r−j},

where ω_{A}^{j}η_{A}^{r−j} is the class introduced in (1.4.6), belongs to H^{0}(C, \underline{ω}^{r} ⊗ Ω_{C}^{1}) ⊗ Sym^{r} H_{dR}^{1}(A) and can thus be viewed, via Proposition 2.5, as an element of the middle step Fil^{r+1} in the Hodge filtration of ϵ_{X}H_{dR}^{2r+1}(X_{r}/F).

2.3. Definition of the cycles. In this section we will assume the Heegner hypothesis 1.9 that was discussed in Section 1.4. As in Section 1.4, fix once and for all a Γ-level structure t_{A} on A, in such a way that t_{A} belongs to A[𝔑].

The datum (A, t_{A}) determines a point P_{A} on C, and a canonical embedding ι_{A} of A^{r} into the fiber in W_{r} above P_{A}. More generally, any pair (φ, A′) ∈ Isog^{𝔑}(A) determines a point P_{A′} on C attached to the pair (A′, φ(t_{A})), and an embedding

ι_{A′} : (A′)^{r}⟶W_{r}

defined over F.

We associate to any (φ, A′) ∈ Isog^{𝔑}(A) a codimension r+1 cycle Υ_{φ} on X_{r} by letting Graph(φ) ⊂ A×A′ denote the graph of φ, and setting

Υ_{φ} := Graph(φ)^{r} ⊂ (A × A′)^{r} ⟶^{≃} (A′)^{r} × A^{r} ⊂ W_{r} × A^{r},

where the last inclusion is induced from the pair (ι_{A′}, id_{A}^{r}). We then set

Δ_{φ} := ϵ_{X}Υ_{φ},

where ϵ_{X} is the idempotent given in equation (2.2.1), viewed as an element of the ring of algebraic correspondences from X_{r} to itself. Note that Δ_{φ} is supported on the fiber π_{r}^{−1}(P_{A′}) of π_{r} above P_{A′} and gives an element in CH^{r+1}(X_{r})_{Q}, the Chow group of codimension r + 1 cycles with rational coefficients.

Remark 2.6. The generalised Heegner cycles Δ_{φ} are all defined over abelian extensions of K. More precisely, if (φ, A′) belongs to Isog_{c}^{𝔑}(A), then the associated cycles can be defined over the compositum of the abelian extension H̃/K over which the isomorphism class of (A, t_{A}) is defined with the ring class field H_{c} of conductor c.

When r = 0, the generalised Heegner cycle Δ_{φ} is a CM point on the modular curve C. In this case, we replace Δ_{φ} by Δ_{φ} − ∞, where ∞ is any cusp, in order to make Δ_{φ} homologically trivial. The same is true when r ≥ 1, by Proposition 2.4 which implies that ϵ_{X}H^{2r+2}(X, Q) = 0. Thus we record the following:

Proposition 2.7. The cycle Δ_{φ} is homologically trivial on X_{r}.

Remark 2.8. Another approach to proving the homological triviality of Δ_{φ}, by deforming these cycles to the fibers supported above the cusps of the modular curve, is described in [Sch]. The approach we have given adapts more readily to the setting of Shimura curves attached to arithmetic subgroups of SL_{2}(R) with compact quotient.

2.4. Relation with Heegner cycles and L-series. This motivational section discusses the relation between generalised Heegner cycles and the more classical Heegner cycles on Kuga-Sato varieties that are studied in [Ne2] and [Zh], as well as the expected relation with derivatives of L-series.

Keeping the same notations as in the previous section, the “traditional” Heegner cycles are codimension r + 1 cycles on the Kuga-Sato variety W_{2r} which are supported on fibers for the natural projection to the modular curve C. These cycles are indexed by elliptic curves with Γ-level structure having endomorphisms by an order in an imaginary quadratic field. More precisely, if A′ is an elliptic curve with endomorphism by the order O_{c} = Z[(d + \sqrt{−d})/2] of conductor c of the imaginary quadratic field K, then we set

Υ_{A′}^{heeg} := graph(\sqrt{−d})^{r} ⊂ (A′ × A′)^{r},
Δ_{A′}^{heeg} := ϵ_{W} (Υ_{A′}^{heeg}).

We will now construct an explicit correspondence from the (4r+1)-dimensional variety X_{2r} to the (2r+1)- dimensional variety W_{2r} which maps generalised Heegner cycles to Heegner cycles.

Let Π = W_{2r} × A^{r}, viewed as a subvariety of W_{2r} × X_{2r} = W_{2r} × W_{2r} × (A^{2})^{r} via the map

(id_{W_{2r}}, id_{W_{2r}}, (id_{A}, \sqrt{−d_{K}})^{r}).

This subvariety induces a correspondence from X_{2r} to W_{2r}, yielding a map on Chow groups:

Φ_{Π} : CH^{2r+1}(X_{2r})_{Q}⟶ CH^{r+1}(W_{2r})_{Q}.

If φ : A⟶A′ is an isogeny of elliptic curves with Γ-level structure, a direct calculation (which will not be used in the sequel, and is therefore left to the reader) shows that the cycles Φ_{Π}(Δ_{φ}) and Δ_{A′}^{heeg} generate the same Q-subspace of CH^{r+1}(W_{2r})_{Q}.

This relation shows that the generalised Heegner cycles carry at least as much information as the classical Heegner cycles on Kuga-Sato varieties studied in [Ne2] and [Zh]. One expects them to enjoy similar relationships with central critical derivatives of Rankin L-series. More precisely, we expect that the Arakelov heights of the generalized Heegner cycles Δ_{φ} should encode the derivatives L′(f, χ^{−1}, 0) where χ are Hecke characters of infinity type (k − 1 − j, 1 + j) with 0 ≤ j ≤ r. The case r = 0 corresponds to the classical Gross-Zagier formula, and the case where r is even and j = r/2 corresponds to the setting that is treated in [Zh]. We expect that there should also be a generalisation of the p-adic result of [Ne2] expressing the p-adic height of generalised Heegner cycles in terms of the derivative in the cyclotomic direction of a two variable p-adic L-function attached to f and χ, at a point which corresponds to the special value L(f, χ^{−1}, 0) and lies in the range of classical interpolation defining this p-adic L-function.

The present article avoids height calculations altogether by focusing instead on the images of generalised Heegner cycles under Abel-Jacobi maps, both complex and p-adic. In the p-adic setting, we will relate these images to the special values of an anticyclotomic p-adic L-function attached to f and K at a point lying outside its range of classical interpolation.

3. p-adic Abel-Jacobi maps

The goal of this section is to compute the images of the generalised Heegner cycles Δ_{φ} under the p-adic Abel-Jacobi map. The resulting formulae of Sections 3.7 and 3.8 are a key ingredient in the proof of our p-adic Gross-Zagier formula. Some of the techniques used in this chapter, particularly those of Sections 3.1–3.4, are drawn from [IS], which treats the case of Heegner cycles on the r-fold product of the universal “fake” elliptic curve over a Shimura curve attached to a quaternion algebra which is ramified at p. This Shimura curve admits an explicit description as a rigid analytic quotient of the p-adic upper half-plane, via the Cerednik-Drinfeld theory of p-adic uniformisation of Shimura curves. The present chapter treats classical modular curves at primes p of good reduction, for which no p-adic uniformisation à la Cerednik-Drinfeld is available. The techniques employed in Section 3.5 onwards therefore differ markedly from those of [IS].

3.1. The étale Abel-Jacobi map. Recall the generalised Heegner cycle Δ_{φ} associated to the pair (φ, A′) ∈ Isog_{c}(A) where φ : (A, t) → (A′, t′) is an isogeny of elliptic curves with Γ-level structure. Let P = P_{A′} be the point of C associated to the pair (A′, t′) and let

X_{P} := π_{r}^{−1}P, X_{r}^{♭} := X_{r} − X_{P} .

Fix any field F over which the pair (X_{r}, Δ_{φ}) is defined, and a rational prime p. Consider the following Gysin sequence in p-adic étale cohomology (cf. Corollary 16.2 of [Mi]). After setting X = X̄_{r}, Z = X̄_{P}, U = X̄_{r}^{♭} and F = Q_{p}(r + 1) in the statement of that corollary (with r replaced by 2r), we obtain the following exact sequence in the category Rep_{F} of continuous p-adic representations of G_{F} = Gal(F̄/F):

(3.1.1) H_{et}^{2r−1}(X̄_{P} , Q_{p})(r)⟶H_{et}^{2r+1}(X̄_{r}, Q_{p})(r+1)⟶H_{et}^{2r+1}(X̄_{r}^{♭}, Q_{p})(r+1)⟶H_{et}^{2r}(X̄_{P} , Q_{p})(r)_{0}⟶0,

where

H_{et}^{2r}(X̄_{P} , Q_{p})(r)_{0} := ker (H_{et}^{2r}(X̄_{P} , Q_{p})(r)⟶H_{et}^{2r+2}(X̄_{r}, Q_{p})(r+1)) .

By applying the projector ϵ_{X} to (3.1.1), we obtain

(3.1.2)  0⟶ϵ_{X}H_{et}^{2r+1}(X̄_{r}, Q_{p})(r+1)⟶ϵ_{X}H_{et}^{2r+1}(X̄_{r}^{♭}, Q_{p})(r+1)⟶ϵ_{X}H_{et}^{2r}(X̄_{P} , Q_{p})(r)⟶0,

where we have used the fact that, when r > 0,

ϵ_{X}H_{et}^{2r−1}(X_{P})(r) = 0, ϵ_{X}H_{et}^{2r}(X_{P})(r)_{0} = ϵ_{X}H_{et}^{2r}(X_{P})(r).

Since Δ = Δ_{φ} is equal to ϵ_{X}Δ_{φ} by definition, its image under the étale cycle class map

cl_{P} : CH^{r}(X_{P})_{Q}(F)⟶H_{et}^{2r}(X̄_{P} , Q_{p})(r)

belongs to ϵ_{X}H_{et}^{2r}(X̄_{P} , Q_{p})(r). Let

cl_{Δ} : Q_{p}⟶ϵ_{X}H_{et}^{2r}(X̄_{P} , Q_{p})(r)

be the map of p-adic representations of G_{F} defined by cl_{Δ}(1) = cl_{P} (Δ), and consider the extension V_{Δ} of Q_{p} by ϵ_{X}H_{et}^{2r+1}(X̄_{r}, Q_{p})(r+1) arising from pullback in the following commutative diagram with exact rows in which the right-most square is cartesian: (3.1.3)

[figure: 0; ϵ_{X}H_{et}^{2r+1}(X̄_{r}, Q_{p})(r+1); V_{Δ}; Q_{p}; _{cl_{Δ}}; 0; 0; ϵ_{X}H_{et}^{2r+1}(X̄_{r}, Q_{p})(r+1); ϵ_{X}H_{et}^{2r+1}(X̄_{r}^{♭}, Q_{p})(r+1); ϵ_{X}H_{et}^{2r}(X̄_{P} , Q_{p})(r); 0.]

Given two objects V′′, V′ in the category Rep_{F}, write

Ext_{F} (V′′, V′) := H^{1}(F, hom(V′′, V′))

for the set of isomorphism classes of extensions

0⟶V′⟶E⟶V′′⟶0.

(Here H^{1}(F, −) denotes continuous Galois cohomology and hom(V′′, V′) is the object of Rep_{F} equipped with the natural action of G_{F}.)

Definition 3.1. The étale Abel-Jacobi map

AJ_{F}^{et} : CH^{r+1}(X_{r})_{0,Q}(F)⟶H^{1}(F, ϵ_{X}H_{et}^{2r+1}(X̄_{r}, Q_{p}(r+1)))

sends the class of the null-homologous codimension-(r+1) cycle Δ to the isomorphism class of the extension V_{Δ} of (3.1.3) in

Ext_{F} (Q_{p}, ϵ_{X}H_{et}^{2r+1}(X̄_{r}, Q_{p})(r+1)) = H^{1}(F, ϵ_{X}H_{et}^{2r+1}(X̄_{r}, Q_{p})(r+1)).

Remark 3.2. Definition 3.1 applies directly to cycle classes in CH^{r+1}(X_{r})_{0,Q}(F) which are represented by a cycle supported on X_{P}. Usually, the map AJ_{F}^{et} is defined on a general cycle Δ by replacing in the diagrams above X_{P} by Δ and X^{♭} by X − Δ, respectively. In this case, one obtains an analogue of the commutative diagram (3.1.3) without the need of applying ϵ_{X}. It can be checked, following the argument that is explained in [Ne2, Prop. II.2.4] that this more general definition, once composed with ϵ_{X}, is compatible with Definition 3.1, which is adapted to our subsequent calculations.

3.2. The comparison isomorphism. The p-adic Abel-Jacobi map arises from the map AJ_{F}^{et} by considering the case where F is a finite extension of Q_{p}. Let O_{F} denote the ring of integers of F and let k be its residue field. We will make the following assumptions on F, which are satisfied in our application:

(1) The extension F is a finite unramified extension of Q_{p}.

(2) The varieties C and X_{r} over F extend to smooth proper models C and X_{r} over O_{F}. If φ belongs to Isog_{c}^{𝔑}(A), and p does not divide Nc Disc(K), then the field F can be taken to be the p-adic completion of the compositum of H̃, the extension of the Hilbert class field of K over which A[𝔑] is defined, with H_{c}, the Hilbert class field of conductor c. By abuse of notation, we will use the same letter σ to denote the p-power Frobenius automorphism of k and its canonical lift to F.

The de Rham cohomology groups H_{dR}^{j}(X_{r}/F), equipped with their σ-semilinear Frobenius endomorphisms and Hodge filtrations, are examples of filtered Frobenius modules. (See [B], [Fo], [I] or [FI] for details concerning the category of these objects.)

The fundamental comparison theorem between p-adic étale cohomology and de Rham cohomology of varieties over p-adic fields relates the p-adic representation H_{et}^{j}(X̄_{r}, Q_{p}) of G_{F} to the filtered Frobenius module H_{dR}^{j}(X_{r}/F). To any continuous p-adic representation V of G_{F} we may associate the F -vector space

D_{cris}(V) := (V ⊗_{Q_{p}} B_{cris})^{G_{F}} ,

where B_{cris} is Fontaine’s ring of cristalline periods over F, which is called the cristalline Dieudonné module attached to V. Recall that a p-adic representation V of G_{F} is said to be cristalline if

dim_{F} D_{cris}(V) = dim_{Q_{p}}(V).

The category of cristalline representations of G_{F} is an abelian tensor subcategory of Rep_{F}. Given objects V_{1} and V_{2} of this category, denote by Ext_{cris}(V_{1}, V_{2}) the group of extensions of V_{2} by V_{1} which are cristalline. The Dieudonné module attached to a cristalline representation V inherits from B_{cris} the structure of a filtered Frobenius module. The following deep theorem will be used to make the p-adic Abel-Jacobi map amenable to computation.

Theorem 3.3 (Faltings). The p-adic representation H_{et}^{2r+1}(X̄_{r}, Q_{p})(r+1) is cristalline, and there is a canonical, functorial isomorphism of filtered Frobenius modules:

D_{cris}(H_{et}^{2r+1}(X̄_{r}, Q_{p})(r+1)) = H_{dR}^{2r+1}(X_{r}/F)(r+1).

Proof. See [Fa], Theorem 5.6, or [T]. □

The comparison theorem will be applied via the following corollary:

Corollary 3.4. The assignment V ↦ D_{cris}(V) induces an isomorphism

(3.2.1)  comp : Ext_{cris}(Q_{p}, H_{et}^{2r+1}(X̄_{r}, Q_{p})(r+1)) ⟶^{∼} Ext_{ffm}(F, H_{dR}^{2r+1}(X_{r}/F)(r+1)).

Proof. The injectivity follows from from the comparison theorem and the fact that the functor D_{cris} is fully faithful, while the surjectivity follows from a comparison with the Bloch-Kato exponential, as in Prop. 1.21 and Cor. 1.22 of [Ne1]. □

3.3. Extensions of filtered Frobenius modules. We will now give a general abstract description of the group of extensions in the category of filtered Frobenius modules.

Let H be a filtered Frobenius module of strictly negative weight, and consider an extension

(3.3.1)  0⟶H ⟶^{i} E ⟶^{ρ} F ⟶0

of filtered Frobenius modules. Let η_{E}^{hol} and η_{E}^{frob} be elements of Fil^{0} E and E^{ϕ^{n}=1} respectively, satisfying

(3.3.2)  ρ(η_{E}^{hol}) = 1,  ρ(η_{E}^{frob}) = 1.

The element

η_{E} := η_{E}^{hol} − η_{E}^{frob}

is in the kernel of ρ and hence can be viewed as an element of H. The lifts η_{E}^{hol} and η_{E}^{frob} are well-defined up to Fil^{0} H and H^{ϕ^{n}=1} respectively. By the assumption on the weight of H, we have H^{ϕ^{n}=1} = 0, and the class of η_{E} in H/ Fil^{0} H does not depend on the choices that were made in (3.3.2). The reader should compare the following proposition with Lemma 2.1 of Section 2 of [IS], which treats the more complicated situation of extensions of filtered Frobenius monodromy modules arising from semistable (and not necessarily cristalline) p-adic representations of G_{F}.

Proposition 3.5. The assignment E ↦ η_{E} yields an isomorphism

Ext_{ffm}(F, H) = H/ Fil^{0} H.

Sketch of proof. The isomorphism E^{ϕ^{n}=1}⟶F induced by ρ determines a canonical vector space splitting of (3.3.1) which preserves the ϕ-module structure of the extension, but need not respect with the filtrations. In other words, the extension (3.3.1) is trivial when viewed as an extension of ϕ-modules. Fix the resulting identification

(3.3.3)  E = H ⊕ F,

so that η_{E}^{frob} is identified with the element (0, 1) of H ⊕ F. We are left with the problem of classifying the filtrations which may arise on the splitting of ϕ-modules (3.3.3). This splitting is compatible with filtrations if and only if η_{E}^{hol} = (h, 1) is such that h belongs to Fil^{0} H (since in this case Fil^{0} E = Fil^{0} H ⊕ F and this equality determines the filtration on E in all degrees). In general, the datum η_{E}^{hol} = (h, 1) completely determines the filtration on E in terms of the filtration on H (since Fil^{0} E = span(Fil^{0} H, η_{E}^{hol})), and (h, 1) and (h′, 1) give rise to the same filtration if and only if h − h′ belongs to Fil^{0} H. □

3.4. The p-adic Abel-Jacobi map. We can now define the p-adic Abel-Jacobi map attached to the p-adic field F introduced in Section 3.2. By Theorem 3.1.1. of [Ne3] (see also [Ni]), the image of CH^{r+1}(X_{r})_{0,Q}(F) by the étale Abel-Jacobi map AJ_{F}^{et} is contained in the subgroup

H_{f}^{1}(F, ϵ_{X}H_{et}^{2r+1}(X̄_{r}, Q_{p})(r+1)) := Ext_{cris}(Q_{p}, ϵ_{X}H_{et}^{2r+1}(X̄_{r}, Q_{p})(r+1))

of H^{1}(F, ϵ_{X}H_{et}^{2r+1}(X̄_{r}, Q_{p})(r+1)) whose elements correspond to cristalline extensions. By Corollary 3.4, this group is identified with Ext_{ffm}(F, ϵ_{X}H_{dR}^{2r+1}(X_{r}/F)(r+1)). Applying Proposition 3.5 to the filtered Frobenius module H = ϵ_{X}H_{dR}^{2r+1}(X_{r}/F)(r+1) which is of weight −1, we find an isomorphism

(3.4.1) J : Ext_{ffm}(F, ϵ_{X}H_{dR}^{2r+1}(X_{r}/F)(r+1))⟶\frac{ϵ_{X}H_{dR}^{2r+1}(X_{r}/F)(r+1)}{Fil^{0} ϵ_{X}H_{dR}^{2r+1}(X_{r}/F)(r+1)} = Fil^{1} ϵ_{X}H_{dR}^{2r+1}(X_{r}/F)(r)^{∨},

where the last identification arises from the Poincaré duality

ϵ_{X}H_{dR}^{2r+1}(X_{r}/F)(r) × ϵ_{X}H_{dR}^{2r+1}(X_{r}/F)(r+1)⟶F,

in which the spaces Fil^{1} ϵ_{X}H_{dR}^{2r+1}(X_{r}/F)(r) and Fil^{0} ϵ_{X}H_{dR}^{2r+1}(X_{r}/F)(r+1) are exact annihilators of each other.

The p-adic Abel-Jacobi map, denoted AJ_{F}, is the diagonal map in the diagram

[figure: CH^{r+1}(X_{r})_{0,Q}(F); ^{AJ{et \atop F}}; _{AJ_{F}}; H_{f}^{1}(F, ϵ_{X}H_{et}^{2r+1}(X̄_{r}, Q_{p})(r+1)); Ext_{cris}(Q_{p}, ϵ_{X}H_{et}^{2r+1}(X̄_{r}, Q_{p})(r+1)); comp; Ext_{ffm}(F, ϵ_{X}H_{dR}^{2r+1}(X_{r})(r+1)); J; (Fil^{1} ϵ_{X}H_{dR}^{2r+1}(X_{r}/F)(r))^{∨},]

where the second vertical isomorphism is given in (3.2.1).

After invoking Proposition 2.5, we can view AJ_{F} as a map

(3.4.2)  AJ_{F} : CH^{r+1}(X_{r})(F)_{0,Q}⟶(S_{r+2}(Γ, F) ⊗ Sym^{r} H_{dR}^{1}(A/F))^{∨}.

Further, applying the comparison isomorphisms to the diagram (3.1.3) gives a corresponding diagram of filtered Frobenius modules:

(3.4.3) [figure: 0; ϵ_{X}H_{dR}^{2r+1}(X_{r}/F)(r+1); D_{Δ}; F; _{cl_{Δ}}; 0; 0; ϵ_{X}H_{dR}^{2r+1}(X_{r}/F)(r+1); ϵ_{X}H_{dR}^{2r+1}(X_{r}^{♭}/F)(r+1); ϵ_{X}H_{dR}^{2r}(X_{P} /F)(r); 0.]

By Proposition 2.4 (and an analogue with C replaced by C − {P }), this diagram can be rewritten as

(3.4.4) [figure: 0; H_{par}^{1}(C, L_{r,r}, ∇)(r+1); D_{Δ}; F; _{cl_{Δ}}; 0; 0; H_{par}^{1}(C, L_{r,r}, ∇)(r+1); H_{par}^{1}(C − {P }, L_{r,r}, ∇)(r+1); L_{r,r}(P)(r); 0.]

The image of the cycle class Δ under the p-adic Abel-Jacobi map is thus described by the class of the extension D_{Δ} in the category of filtered Frobenius modules.

3.5. de Rham cohomology over p-adic fields. In this section we give an explicit description of the action of the Frobenius operator on

ϵ_{X}H_{dR}^{2r+1}(X_{r}/F) = H_{par}^{1}(C, L_{r,r}, ∇)

in terms of L_{r,r}-valued rigid analytic differentials on appropriate subsets of the curve C. The reader is referred to [Col1] and [Col2] for more details on the concepts and definitions discussed below.

Viewing C as a rigid analytic space over F, let O_{C}^{rig} denote the sheaf of rigid analytic functions on C and let L_{r,r}^{rig} denote the rigid analytic coherent sheaf on C associated to L_{r,r}.

We will now define certain basic affinoid subsets of C for the rigid analytic topology. For this, recall from Section 3.2 that C is a smooth proper model of C over Spec(O_{F}). Write C_{k} := C ×_{O_{F}} k, and let

red_{p} : C(C_{p})⟶C_{k}(k̄)

denote the natural reduction map.

Let P_{1}, …, P_{t} be any collection of points on C(F) which map to distinct points of C_{k}(k) under red_{p} and contain all the cusps of C. Recall that the residue disc attached to P_{j}, denoted D(P_{j}), is the set of points of C(C_{p}) which have the same image as P_{j} under red_{p}. Let

A = C(C_{p}) − D(P_{1}) − ⋯ − D(P_{t}).

Because the P_{j} reduce to smooth points of C_{k}(k), the residue discs D(P_{j}) are conformal to the open unit disc U ⊂ C_{p} consisting of z ∈ C_{p} with |z| < 1. For each j = 1, …, t, fix an isomorphism h_{j} : D(P_{j})⟶U sending P_{j} to 0. Given a rational number r_{j} < 1, we then let

D[P_{j}, r_{j}] = {z ∈ D(P_{j}) such that |h_{j}(z)| ≤ r_{j}}

denote the “closed disc of radius r_{j} in D(P_{j})”. Finally, fixing a collection of rational numbers r_{1}, …, r_{t} with 0 < r_{j} < 1, we write

W = C(C_{p}) − D[P_{1}, r_{1}] − ⋯ − D[P_{t}, r_{t}]
= A ∪ V_{1} ∪ ⋯ ∪ V_{t},

where

V_{j} := V(P_{j}, r_{j}, 1) := {z ∈ D(P_{j}) such that r_{j} < |h_{j}(z)| < 1}.

Define the positive orientation of the annulus V_{j} by choosing the subset {z ∈ D(P_{j}) such that |h_{j}(z)| ≤ r_{j}} of its complement.

The set A is an example of an affinoid subset of C(C_{p}) with good reduction, while the set W is an example of a wide open neighborhood of the affinoid A. The set V_{j} is called a wide open annulus around the point P_{j}. The wide open space W is thus obtained by adjoining to A a finite union of open annuli about the boundaries of the deleted residue discs. For general definitions and a more systematic discussion of these concepts, see for example Sections II and III of [Col2].

Because W is contained in C^{0}(C_{p}), the Gauss-Manin connection (1.1.3) gives rise to a rigid analytic connection

∇ : L_{r,r}^{rig}⟶L_{r,r}^{rig} ⊗ Ω_{W}^{1}.

The de Rham cohomology H_{dR}^{1}(W, L_{r,r}^{rig}, ∇) is defined to be the quotient

H_{dR}^{1}(W, L_{r,r}^{rig}, ∇) := \frac{L_{r,r}^{rig}(W) ⊗ Ω_{W}^{1}}{∇L_{r,r}^{rig}(W)} .

A meromorphic L_{r,r}-valued differential on C which is regular on C − {P_{1}, …, P_{t}} can be viewed as a rigid section of L_{r,r} ⊗ Ω_{C}^{1} over W. In this way one obtains by restriction a natural map from the algebraic de Rham cohomology over C_{p} to the rigid de Rham cohomology.

Theorem 3.6. The natural restriction map

H_{dR}^{1}(C − {P_{1}, …, P_{t}}, L_{r,r}, ∇)⟶H_{dR}^{1}(W, L_{r,r}^{rig}, ∇)

is an isomorphism.

Proof. In the case r = 0, this is Theorem 4.2 of [Col2]. The proof in the general case follows from a similar argument, as explained in the proof of Proposition 10.3 of [Col3]. □

A set W′ of the form

W′ = C(C_{p}) − D[P_{1}, r′_{1}] − ⋯ − D[P_{t}, r′_{t}], with r_{j} < r′_{j} < 1

is called a wide open neighborhood of the affinoid A in W. The following is an immediate corollary of Theorem 3.6.

Corollary 3.7. Let W′ be any wide open neighborhood of A in W. The natural map

res_{W,W′} : H_{dR}^{1}(W, L_{r,r}^{rig}, ∇)⟶H_{dR}^{1}(W′, L_{r,r}^{rig}, ∇)

induced by restriction is an isomorphism.

We want to describe the image of H_{dR}^{1}(C, L_{r,r}, ∇) in H_{dR}^{1}(W, L_{r,r}^{rig}, ∇). For this, we recall the notion of the p-adic annular residue

res_{V_{j}}(ω) ∈ (H^{0}(V_{j}, L_{r,r}^{rig})^{∇=0})^{∨}

of a L_{r,r}^{rig}-valued one-differential form ω on W. It is defined by the formula

res_{V_{j}}(ω)(α) = res_{V_{j}}⟨α, ω⟩, for all α ∈ H^{0}(V_{j}, L_{r,r}^{rig})^{∇=0},

where the residue on the right hand side is the usual p-adic annular residue of the rigid analytic one-form ⟨α, ω⟩ on the oriented annulus V_{j}, as it is defined in Section II of [Col2] for example.

By Proposition 3.1.2 of [Ka3], the sheaf L_{r,r} admits a basis of horizontal sections on each non-cuspidal residue disc D(P_{j}), so that the target of the residue map on the corresponding annulus is identified with

(H^{0}(V_{j}, L_{r,r}^{rig})^{∇=0})^{∨} = (H^{0}(D(P_{j}), L_{r,r})^{∇=0})^{∨} = L_{r,r}(P_{j})^{∨} = L_{r,r}(P_{j}),

where the last identification arises from the self-duality on L_{r,r}(P_{j}). We will always view the residue map on a non-cuspidal residue disc as taking values on L_{r,r}(P_{j}), so that for all α ∈ L_{r,r}(P_{j}) one has

⟨α, res_{V_{j}}(ω)⟩ = res_{V_{j}}⟨α^{∇}, ω⟩,

where α^{∇} is the unique horizontal section on D(P_{j}) satisfying α^{∇}(P_{j}) = α.

On the cuspidal residue disc of the cusp P attached to the pair (Tate(q), t), the space of horizontal sections of L_{r} is one-dimensional and generated by the local section ξ_{can}^{r}. One therefore has

res_{V_{j}} ((∑_{j=0}^{r} a_{j}(q)ω_{can}^{j}ξ_{can}^{r−j}) \frac{dq}{q} ) (bξ_{can}^{r}) = res_{q=0} (ba_{r}(q)\frac{dq}{q} ) = ba_{r}(0).

Note that if ω is any global section of L_{r,r} ⊗ Ω_{C}^{1} over C − {P_{1}, …, P_{t}}, it can also be viewed as a rigid section over W, and

(3.5.1)  res_{V_{j}} ω = res_{P_{j}} ω.

If P_{j} is not a cusp, the residue res_{P_{j}} ω that appears on the right of this formula satisfies

⟨G(P_{j}), res_{P_{j}} ω⟩ = res_{P_{j}}⟨G, ω⟩.

In this formula, G can be taken to be any regular (not necessarily horizontal) section of L_{r,r} over D(P_{j}), and the residue on the right is the residue at P_{j} of the differential ⟨G, ω⟩ on D(P_{j}) − {P_{j}}.

The following rigid-analytic analogue of the classical residue theorem for meromorphic differentials on curves (cf. for example [Col2]) will play an important role in the calculations of the next section.

Theorem 3.8. If ω ∈ Ω_{W}^{1} is a rigid analytic one-form on W, then

∑_{j=1}^{t} res_{V_{j}} ω = 0.

Proposition 3.9. A class κ ∈ H_{dR}^{1}(W, L_{r,r}^{rig}, ∇) represented by an L_{r,r}^{rig}-valued differential form ω belongs to the natural image of H_{par}^{1}(C, L_{r,r}, ∇) under restriction if and only if

res_{V_{j}}(ω) = 0, for j = 1, …, t.

Proof. The Gysin exact sequence applied to the cohomology of the pair of rigid spaces W ⊂ C^{0} shows that

H_{dR}^{1}(C, L_{r,r}, ∇) = {ω s.t. res_{V_{j}}(ω) = 0 for all non-cuspidal annuli V_{j}} .

On the other hand, the definition of H_{par}^{1}(C, L_{r,r}, ∇) shows that this space is identified with the space of classes in H_{dR}^{1}(C, L_{r,r}, ∇) represented by L_{r,r}-valued differentials ω satisfying

res_{V_{j}}(ω) = 0, for all cuspidal annuli V_{j}.

The result follows. □

Let κ_{1}, κ_{2} be classes in H_{par}^{1}(C, L_{r,r}, ∇) and let ω_{1}, ω_{2} be rigid analytic sections of L_{r,r}^{rig} ⊗ Ω_{C}^{1} over W representing them. The fact that res_{V_{j}}(ω_{1}) = 0 on all the annuli V_{j} ⊂ W allows us to find an analytic solution F_{ω_{1},j} on V_{j} to the equation

∇F_{ω_{1},j} = ω_{1},

which is well-defined up to horizontal sections of L_{r,r}^{rig} over V_{j}. Such an F_{ω_{1},j} is called a local primitive of ω_{1} on V_{j}. Note that the expression res_{V_{j}}⟨F_{ω_{1},j}, ω_{2}⟩ does not depend on the choice of the local primitive F_{ω_{1},j}, since ω_{2} is of the second kind.

The following proposition expresses the Poincaré duality on H_{par}^{1}(C, L_{r,r}, ∇) in terms of the residues of rigid L_{r,r}-valued forms on W.

Proposition 3.10. For all κ_{1}, κ_{2} ∈ H_{par}^{1}(C, L_{r,r}, ∇),

⟨κ_{1}, κ_{2}⟩ = ∑_{j=1}^{t} res_{V_{j}}⟨F_{ω_{1},j}, ω_{2}⟩,

where ω_{1}, ω_{2} ∈ H_{dR}^{1}(W, L_{r,r}^{rig}, ∇) are representatives for κ_{1} and κ_{2} and F_{ω_{1},j} is any local primitive for ω_{1} on V_{j}.

Proof. This follows from Lemma 7.1 of [Col3] combined with equation (3.5.1) comparing the rigid analytic and algebraic residue maps. □

Theorem 3.6 will now be used to give an explicit description of the action of the Frobenius operator on the algebraic de Rham cohomology. Since the points P_{1}, …, P_{t} are defined over F, the points P̃_{j} := red_{p}(P_{j}) are defined over k and the curve U_{k} := C_{k} − {P̃_{1}, …, P̃_{t}} is a smooth affine open subset of C_{k}. As before, let σ denote the Frobenius automorphism of k which sends x to x^{p}, and let U_{k}^{σ} = U_{k} ×_{σ} k. There is a canonical morphism ϕ : U_{k}⟶U_{k}^{σ} characterised by

ϕ^{∗}f^{σ} = f^{p}, for all f ∈ O_{C_{k}}(U_{k}).

Definition 3.11. A morphism

ϕ_{A} : A⟶A^{σ}

which lifts the canonical Frobenius morphism U_{k}⟶U_{k}^{σ} to characteristic 0 is called a lifting of Frobenius for the affinoid A.

A Frobenius lifting always exists under our hypotheses (cf. Corollary 1.1a of [Col1]). Assume from now on that the set {P̃_{1}, …, P̃_{t}} is stable under ϕ, so that A^{σ} = A.

Definition 3.12. A Frobenius neighbourhood of A in W is a pair (W′, ϕ), where A ⊂ W′ ⊂ W is a wide open neighborhood of A in W and ϕ : W′⟶W is a morphism whose restriction to A is a lifting of Frobenius in the sense of Definition 3.11.

Definition 3.13. An overconvergent Frobenius isocrystal on W is a triple (L, ϕ, Fr), where (1) L is a rigid analytic coherent sheaf on W equipped with a rigid analytic integrable connection

∇ : L⟶L ⊗ Ω_{W}^{1};

(2) (W′, ϕ) is a Frobenius neighborhood of A in W;

(3) Fr is a horizontal morphism

Fr : ϕ^{∗}L⟶L|_{W′}.

The condition that Fr be horizontal amounts to requiring that the diagram

\xymatrix{ϕ^{∗}L \ar[r]^{∇} \ar[d]_{Fr} & ϕ^{∗}L ⊗ Ω_{W′}^{1} \ar[d]^{Fr ⊗id} \\ L \ar[r]^{∇} & L ⊗ Ω_{W′}^{1}}

be commutative.

Given a Frobenius neighborhood (W′, ϕ) of A in W, the canonical functorial action of a lifting of Frobenius on the relative de Rham cohomology H_{dR}^{2r}(X_{r}/C) is compatible with the Gauss-Manin connection and gives rise to a horizontal morphism Fr : ϕ^{∗}L_{r,r}^{rig}⟶L_{r,r}^{rig}|_{W′}. In this way, the triple (L_{r,r}^{rig}_{|W}, ϕ, Fr) is equipped with the structure of an overconvergent Frobenius isocrystal.

The action of the p-power Frobenius operator (denoted by the letter Φ_{0}, to distinguish it from the lifting ϕ of Frobenius on the curve C) on H_{dR}^{1}(W, L_{r,r}^{rig}, ∇) is then given by the sequence of maps:

H_{dR}^{1}(W, L_{r,r}^{rig}, ∇) ⟶^{ϕ^{∗}} H_{dR}^{1}(W′, ϕ^{∗}L_{r,r}^{rig}, ∇) ⟶^{Fr} H_{dR}^{1}(W′, L_{r,r}^{rig}, ∇) ⟶^{∼} H_{dR}^{1}(W, L_{r,r}^{rig}, ∇),

where the last map is the inverse of the restriction res_{W,W′} which is an isomorphism by Corollary 3.7. (Cf. the discussion preceding Th. 10.1 of [Col3], or the more detailed discussion in [CI].)

Notice that the operator Φ_{0} acting on the group H_{dR}^{1}(W, L_{r,r}^{rig}, ∇) preserves the natural images of H_{dR}^{1}(C, L_{r,r}, ∇) and of H_{par}^{1}(C, L_{r,r}, ∇). (This follows from Proposition 3.9 for example.) The map Φ_{0} on H_{par}^{1}(C, L_{r,r}, ∇) agrees with the Frobenius endomorphism on ϵ_{X}H_{dR}^{2r+1}(X_{r}/F) via the identification H_{par}^{1}(C, L_{r,r}, ∇) = ϵ_{X}H_{dR}^{2r+1}(X_{r}/F). It is σ-semilinear. In order to work with an F -linear endomorphism, we set

Φ = Φ_{0}^{n}, where n = [F : Q_{p}].

By abuse of notation, we will also denote by Φ the Frobenius endomorphism acting on the space H_{la}^{0}(C, L_{r})^{∇} of locally analytic horizontal sections of L_{r} over C, as it is described in the paragraph preceding Thm. 10.1 of [Col3].

A similar discussion applies of course when L_{r,r} is replaced by L_{r}, and the symbol Φ will also be used to denote the F -linear Frobenius endomorphism acting on H_{par}^{1}(C, L_{r}, ∇) and H_{la}^{0}(C, L_{r})^{∇}.

3.6. The Coleman primitive.

Lemma 3.14. Let ω be a global (rigid) section of the sheaf \underline{ω}^{r} ⊗ Ω_{C}^{1} over C, and let [ω] ∈ H_{par}^{1}(C, L_{r}, ∇) be its associated cohomology class. There exists a polynomial P ∈ F [x] satisfying

(1) P (Φ)([ω]) = 0.

(2) The map P (Φ) is an isomorphism on H_{la}^{0}(C, L_{r})^{∇}, and P (1) ≠ 0.

Proof. This follows from the ideas explained in Section 11 of [Col3]. (Cf. in particular the argument following Lemma 11.1 of loc.cit.) One can use the fact that the eigenvalues of Φ acting on H_{dR}^{1}(C, L_{r}, ∇) and on any (finite-dimensional) Φ-stable subspace of H_{la}^{0}(C, L_{r})^{∇} differ, since they have complex absolute values p^{\frac{r+1}{2}} and p^{\frac{r}{2}} respectively. □

Theorem 3.15 (Coleman). Let ω be a global section of the sheaf \underline{ω}^{r} ⊗ Ω_{C}^{1} over C. Choose a polynomial P satisfying the properties of Lemma 3.14, and let d be its degree. There exists a locally analytic section F_{ω} of L_{r} over C satisfying the following conditions:

(1) ∇F_{ω} = ω;

(2) P (Φ)(F_{ω}) is a rigid analytic section of L_{r} on some wide open neighborhood W′ of A in W satisfying

ϕ^{n}(W′) ⊂ W, for all n ≤ d. The locally analytic section F_{ω} is called the Coleman primitive of ω on C.

Proof. See Theorem 10.1 of [Col3]. Note that our setting, where p is assumed to not divide the level of the modular curve C, differs from the semistable reduction case considered in [Col3]. In fact it is simpler, and the assumptions that are required for Theorem 10.1 of loc.cit., such as the “regular singular annuli” assumption on the cuspidal annuli, are satisfied a fortiori in the setting of Theorem 3.15. Note also that Theorem 10.1 as stated produces a locally analytic primitive on each wide open W, but expressing C as a finite union of wide opens and gluing the different primitives (which, by their uniqueness, agree on the overlaps) leads to a locally analytic primitive on all of C. The uniqueness clause in the definition of the Coleman primitive also implies that F_{ω} is defined over the field F over which ω is defined. □

Remark 3.16. The definition of F_{ω} depends a priori on several choices: the choice of an affinoid A in C, a lifting of Frobenius to A, a Frobenius neighborhood W′ of A in W and the polynomial P. It can be shown that the Coleman primitive does not depend on these choices, and therefore the Coleman primitives on a covering of C by affinoid regions can be pieced together to give a locally analytic section of L_{r} over C which is well-defined up to global rigid analytic horizontal sections of L_{r} over C. This latter space is trivial when r > 0 and is the space of constant functions on C when r = 0. (Cf. Proposition 5.1 of [Col3].)

Remark 3.17. It can be shown that the Coleman primitive F_{ω} is in fact analytic on each residue disc D(P) associated to any point P of C(Q_{p}^{unram}).

3.7. p-adic integration and the p-adic Abel-Jacobi map. The following is one of the main results of this chapter.

Proposition 3.18. Let Δ_{φ} be a generalised Heegner cycle attached to an isogeny of ordinary pairs φ : (A, t)⟶(A′, t′), and let P_{A′} be the point of C attached to (A′, t′). Then

AJ_{F} (Δ_{φ})(ω_{f} ∧ α) = ⟨F_{f}(P_{A′}) ∧ α, cl_{P_{A′}} (Δ_{φ})⟩,

where the pairing on the right is the natural one on L_{r,r}(P_{A′}), and F_{f} is the Coleman primitive of ω_{f} ∈ H^{0}(C, \underline{ω}^{r} ⊗ Ω_{C}^{1}).

Proof. In order to ease notations, we drop the index φ in this proof, by setting Δ = Δ_{φ}, and write P = P′_{A} and U = C − {P }. By definition of the p-adic Abel-Jacobi map, we have

AJ_{F} (Δ)(ω_{f} ∧ α) = ⟨ω_{f} ∧ α, η_{Δ}⟩,

where the class η_{Δ} represents the extension D_{Δ} of (3.4.4) following the recipe given in Section 3.3. We may write

η_{Δ} = η_{Δ}^{hol} − η_{Δ}^{frob},

where

(1) The cohomology class η_{Δ}^{hol} is represented by a section of L_{r,r} ⊗ Ω_{C}^{1}(log Z_{N}) over U having residue 0 at the cusps and a simple pole at P with residue equal to cl_{P} (Δ). By abuse of notation, we will use the same symbol η_{Δ}^{hol} to denote the associated L_{r,r}-valued differential on C. If P_{1}, …, P_{t} were chosen in such a way that P_{1} = P, and G_{j} is any rigid analytic section of L_{r,r}^{rig} over D(P_{j}), then by (3.5.1), for all non-cuspidal annuli V_{j},

(3.7.1)  res_{V_{1}}⟨G_{1}, η_{Δ}^{hol}⟩ = ⟨G_{1}(P), cl_{P} (Δ)⟩,  res_{V_{j}}⟨G_{j}, η_{Δ}^{hol}⟩ = 0 for j ≥ 2.

If V_{j} is a cuspidal annulus, then we at least have

(3.7.2)  res_{V_{j}}⟨F_{f,j} ∧ α, η_{Δ}^{hol}⟩ = 0,

where F_{f,j} is a local primitive of ω_{f} on V_{j}. To see this, use the fact that η_{Δ}^{hol} has residue 0 along V_{j} to write η_{Δ}^{hol} = ∇H_{Δ} for some section of L_{r,r}^{rig} over V_{j}, and observe that

0 = res_{V_{j}} d⟨F_{f,j} ∧ α, H_{Δ}⟩ = res_{V_{j}}(⟨ω_{f}, H_{Δ}⟩ + ⟨F_{f,j} ∧ α, η_{Δ}^{hol}⟩) = res_{V_{j}}⟨F_{f,j} ∧ α, η_{Δ}^{hol}⟩.

(2) The differential η_{Δ}^{frob} is a section of L_{r,r}^{rig} ⊗ Ω_{C}^{1} over W, chosen so that it satisfies
(3.7.3)  Φη_{Δ}^{frob} = η_{Δ}^{frob} + ∇G,

for some rigid section G of L_{r,r}^{rig} over W′, and of course

(3.7.4)  res_{V_{1}}⟨G_{1}, η_{Δ}^{Frob}⟩ = ⟨G_{1}(P), cl_{P} (Δ)⟩.

By Proposition 3.10, the Poincaré pairing between H_{dR}^{1}(C, L_{r,r}(r), ∇) and H_{dR}^{1}(C, L_{r,r}(r+1), ∇) is given by the formula

(3.7.5)  ⟨ω_{f} ∧ α, η_{Δ}⟩ = ∑_{j=1}^{t} res_{V_{j}}⟨F_{f,j} ∧ α, η_{Δ}⟩

(3.7.6)  = (∑_{j=1}^{t} res_{V_{j}}⟨F_{f,j} ∧ α, η_{Δ}^{hol}⟩) − (∑_{j=1}^{t} res_{V_{j}}⟨F_{f,j} ∧ α, η_{Δ}^{frob}⟩) ,

where the sum is taken over the t annuli V_{j} in W − A, and F_{f,j} is an analytic primitive of ω_{f} on the residue disc D(P_{j}). Note that if ω^{∇} is any horizontal section of L_{r,r} on D(P_{j}), the residue of the differential ⟨ω^{∇}, η_{Δ}⟩ on the annulus V_{j} is zero, and therefore the expression on the right of (3.7.5) is independent of the choice of local primitives on each residue disc. The same is not true for either of the sums that appear on the right of (3.7.6), since the differentials η_{Δ}^{hol} and η_{Δ}^{Frob} each have non-zero residue along the annulus V_{1}.

In order to compute each of the terms appearing in (3.7.6) individually, we need to make a “coherent” choice of local primitives. This is done by fixing a Coleman primitive F_{f} of ω_{f}. Once this choice is made, the two terms appearing in (3.7.6) are controlled in the following two lemmas.

Lemma 3.19. If F_{f,j} is any choice of local primitives of ω_{f} on each residue disc D(P_{j}), then

∑_{j=1}^{t} res_{V_{j}}⟨F_{f,j} ∧ α, η_{Δ}^{hol}⟩ = ⟨F_{f,1}(P_{A′}) ∧ α, cl_{P_{A′}} (Δ)⟩.

Proof. Since the local primitive F_{f,j} ∧ α is analytic on the residue disc D(P_{j}), and since η̃_{Δ}^{hol} has 0 residue on V_{j} when j ≥ 2, it follows from (3.7.1) and (3.7.2) that

∑_{j=1}^{t} res_{V_{j}}⟨F_{f,j} ∧ α, η_{Δ}^{hol}⟩ = res_{V_{1}}⟨F_{f,1} ∧ α, η_{Δ}^{hol}⟩ = ⟨F_{f}(P_{A′}) ∧ α, cl_{P_{A′}} (Δ)⟩.

The lemma follows. □

Lemma 3.20. Let F_{f} be the Coleman primitive of ω_{f} on C. Then

(3.7.7)  ∑_{j=1}^{t} res_{V_{j}}⟨F_{f} ∧ α, η_{Δ}^{frob}⟩ = 0.

Proof. We begin by noting that for each j = 1, …, t,

res_{V_{j}}⟨F_{f} ∧ α, η_{Δ}^{Frob}⟩ = res_{V_{j}}⟨ΦF_{f} ∧ α, Φη_{Δ}^{Frob}⟩
(3.7.8)  = res_{V_{j}}⟨ΦF_{f} ∧ α, η_{Δ}^{Frob}⟩ + res_{V_{j}}⟨ΦF_{f} ∧ α, ∇G⟩,

where G is the rigid analytic section of L_{r,r} over W′ given by (3.7.3). The fact that Φ is horizontal for the Gauss-Manin connection (combined with the Leibniz rule) shows that

d⟨ΦF_{f} ∧ α, G⟩ = ⟨ΦF_{f} ∧ α, ∇G⟩ + ⟨Φω_{f} ∧ α, G⟩.

In particular, the expression appearing on the right is exact on each annulus V_{j}, and therefore

∑_{j=1}^{t} res_{V_{j}}⟨ΦF_{f} ∧ α, ∇G⟩ = − ∑_{j=1}^{t} res_{V_{j}}⟨Φω_{f} ∧ α, G⟩
= 0,

where the last vanishing follows from the rigid analytic residue theorem (Theorem 3.8), in light of the fact that ⟨Φω_{f} ∧ α, G⟩ belongs to Ω_{W′}^{1}. Hence by summing equation (3.7.8) over j = 1, …, t, we get

∑_{j=1}^{t} res_{V_{j}}⟨F_{f} ∧ α, η_{Δ}^{Frob}⟩ = ∑_{j=1}^{t} res_{V_{j}}⟨ΦF_{f} ∧ α, η_{Δ}^{Frob}⟩.

More generally, if L is any polynomial in F [x], we get

L(1) ∑_{j=1}^{t} res_{V_{j}}⟨F_{f} ∧ α, η_{Δ}^{Frob}⟩ = ∑_{j=1}^{t} res_{V_{j}}⟨L(Φ)F_{f} ∧ α, η_{Δ}^{Frob}⟩.

Now, choosing the polynomial L(x) = P (x) as in Lemma 3.14 preceding the definition of the Coleman primitive, we get

L(1) ∑_{j=1}^{t} res_{V_{j}}⟨F_{f} ∧ α, η_{Δ}^{Frob}⟩ = ∑_{j=1}^{t} res_{V_{j}}⟨L(Φ)F_{f} ∧ α, η_{Δ}^{Frob}⟩ = 0,

where the vanishing follows by noting that L(Φ)F_{f} ∧ α is a rigid analytic section of L_{r,r} over W′ and applying Theorem 3.8 once again. Lemma 3.20 now follows from the fact that L(1) ≠ 0. □

The proof of Proposition 3.18 now follows from (3.7.6) combined with Lemmas 3.19 and 3.20, which show that

AJ_{F} (Δ_{φ})(ω_{f} ∧ α) = ⟨ω_{f} ∧ α, η_{Δ}⟩ = ⟨F_{f}(P_{A′}) ∧ α, cl_{P_{A′}} (Δ)⟩

when F_{f} is a Coleman primitive for ω_{f}. □

Proposition 3.21. With the same notations as in Proposition 3.18,

AJ_{F} (Δ_{φ})(ω_{f} ∧ α) = ⟨φ^{∗}F_{f}(P_{A′}), α⟩_{A},

where the pairing ⟨ , ⟩_{A} on the right is the natural one on Sym^{r} H_{dR}^{1}(A/F).

Proof. Let

ϱ := (φ^{r}, id^{r}) : A^{r}⟶Υ_{φ} ⊂ (A′)^{r} × A^{r}.

Note that

ϱ^{∗}(F_{f}(P_{A′}) ∧ α) = φ^{∗}(F_{f}(P_{A′})) ∧ α, ϱ([A^{r}]) = cl_{P_{A′}} (Υ_{φ}),

where [A^{r}] ∈ H_{dR}^{0}(A^{r}/F) is the fundamental class associated to the variety A^{r}. Let

⟨ , ⟩_{A,j} : H_{dR}^{2r−j}(A^{r}/F) × H_{dR}^{j}(A^{r}/F)⟶H^{2r}(A^{r}/F) = F

denote the Poincaré pairing, so that the restriction of ⟨ , ⟩_{A,r} to Sym^{r} H_{dR}^{1}(A/F) ⊂ H_{dR}^{r}(A/F) agrees with ⟨ , ⟩_{A}. Observe that

(3.7.9)  ⟨F_{f}(P_{A′}) ∧ α, cl_{P_{A′}} (Δ_{φ})⟩ = ⟨F_{f}(P_{A′}) ∧ α, cl_{P_{A′}} (Υ_{φ})⟩ = ⟨F_{f}(P_{A′}) ∧ α, ϱ([A^{r}])⟩.

The functoriality properties of the Poincaré pairing imply that

⟨F_{f}(P_{A′}) ∧ α, ϱ([A^{r}])⟩ = ⟨ϱ^{∗}(F_{f}(P_{A′}) ∧ α), [A^{r}]⟩_{A,0}
(3.7.10)  = ⟨φ^{∗}(F_{f}(P_{A′})) ∧ α, [A^{r}]⟩_{A,0} = ⟨φ^{∗}(F_{f}(P_{A′})), α⟩_{A}.

Proposition 3.21 follows by combining Proposition 3.18 with (3.7.9) and (3.7.10). □

Let {P̄_{1}, …, P̄_{t}} be the set of supersingular points of C_{k}, and let P_{j} ∈ C(F) be an arbitrary lift of P̄_{j} under the reduction map. The residue discs D(P_{j}) are called the supersingular discs of C and the complement A := C^{ord} is called the ordinary locus of C. A locally analytic p-adic modular form of weight k is a locally analytic section of ω^{k} over C^{ord}. Following equation (1.1.1), a modular form G of this type can also be viewed as a function on ordinary triples of generalised elliptic curves (E, t, ω)_{/R}, where R is a p-adic ring of finite type over Z_{p}, satisfying

G(E, t, λω) = λ^{−k}G(E, t, ω), for all λ ∈ R^{×}.

Following Chapter VII of [DR], in particular Corollaire 2.2, the formal completion along a cusp of a suitable cuspidal p-adic neighborhood D ≃ Spec(R) in C^{ord} can be identified with Spf(Z[[q^{1/d}]]), for Z finite unramified over Z_{p} and d | N, in such a way that the universal object over D pulls back to Tate(q), equipped with a suitable level structure. By an abuse of notation, we will denote by G(Tate(q), t, ω_{can}) the q-expansion obtained by evaluating G at a generalised marked elliptic curve corresponding to (Tate(q), t, ω_{can}) via the above identifications.

For 0 ≤ j ≤ r, let G_{j} denote the “j-th component” of the Coleman primitive F_{f}, defined (as a function on ordinary triples) by the rule

G_{j}(E, t, ω) := ⟨F (E, t), ω^{j}η^{r−j}⟩,

where η is the generator of the unit root subspace of H_{dR}^{1}(E/R), normalised so that ⟨ω, η⟩ = 1. The rule G_{j} thus defined satisfies

G_{j}(E, t, λω) = λ^{2j−r}G_{j}(E, t, ω), for all λ ∈ R^{×},

and therefore defines a locally analytic p-adic modular form of weight r − 2j.

The next lemma expresses the Abel-Jacobi images of the cycles Δ_{φ} in terms of the modular forms G_{j}.

Lemma 3.22. Let

φ : (A, t, ω)⟶(A′, t′, ω′)

be an isogeny of ordinary marked elliptic curves of degree d_{φ} = deg(φ), and let Δ_{φ} be the associated generalised Heegner cycle on X_{r}. Then

AJ_{F} (Δ_{φ})(ω_{f} ∧ ω^{j}η^{r−j}) = d_{φ}^{j}G_{j}(A′, t′, ω′).

Proof. By Proposition 3.21,

(3.7.11)  AJ_{F} (Δ_{φ})(ω_{f} ∧ ω^{j}η^{r−j}) = ⟨φ^{∗}F_{f}(A′, t′), ω^{j}η^{r−j}⟩_{A}.

Since ⟨φ^{∗}ω′, φ^{∗}η′⟩ = d_{φ}, we have

(3.7.12)  φ^{∗}(η′) = d_{φ}η.

Hence

⟨φ^{∗}F_{f}(A′, t′), ω^{j}η^{r−j}⟩_{A} = d_{φ}^{j−r}⟨φ^{∗}F_{f}(A′, t′), φ^{∗}((ω′)^{j}(η′)^{r−j})⟩_{A}
= d_{φ}^{j}⟨F_{f}(A′, t′), (ω′)^{j}(η′)^{r−j}⟩_{A′}
= d_{φ}^{j}G_{j}(A′, t′, ω′).
□

3.8. Calculation of the Coleman primitive. We now turn to the explicit calculation of the Coleman primitive F_{f} of the regular L_{r}^{rig}-valued differential ω_{f}, or rather, of its components G_{j}. In order to do this, we begin by introducing an operator V U − UV on locally analytic p-adic modular forms which plays the role of the operator P (Φ) in Theorem 3.15 defining the Coleman primitive, in the sense that it maps the locally analytic forms G_{j} to genuine p-adic modular forms in the sense of Section 1.3. As a consequence of the use of this operator, it will be possible to resort to q-expansions in our calculation of Coleman primitive (cf. the proof of Proposition 3.24).

We recall the definition of the operators U and V (as they are described in [Se] for example). Given an ordinary triple (E, t, ω), let

φ_{j}^{(p)} : (E, t, ω)⟶(E_{j}, t_{j}, ω_{j}), j = 0, 1, …, p

denote the distinct p-isogenies on E, ordered in such a way that φ_{0}^{(p)} is the distinguished p-isogeny whose kernel is the canonical subgroup of E. For instance, when (E, t, ω) = (Tate(q), ζ_{N}, ω_{can}), the canonical subgroup is μ_{p} and we can take

(3.8.1)  (E_{0}, t_{0}, ω_{0}) = (Tate(q^{p}), ζ_{N}^{p}, \frac{1}{p}ω_{can}) ,  (E_{j}, t_{j}, ω_{j}) = (Tate(q^{1/p}ζ_{p}^{j}), ζ_{N}, ω_{can}).

The Hecke operators U and V are defined by setting

(G|U)(E, t, ω) := G(U(E, t, ω)), (G|V)(E, t, ω) := G(V (E, t, ω)),

where

U(E, t, ω) := \frac{1}{p} ∑_{j=1}^{p}(E_{j}, t_{j}, ω_{j}), V (E, t, ω) := (E_{0}, \frac{1}{p}t_{0}, pω_{0}).

These operators are related to the usual Hecke operator T_{p} by the rule

T_{p} = U + \frac{1}{p}[p]V,

where [p] denotes the isogeny given by multiplication by p. In particular,

(3.8.2)  V U − UV = 1 − T_{p}V + \frac{1}{p}[p]V^{2}.

The diamond operator ⟨a⟩ attached to a ∈ (Z/NZ)^{×} is defined on locally analytic p-adic modular forms by the rule

G|⟨a⟩(E, t, ω) = G(E, a^{−1}t, ω).

Given a locally analytic p-adic modular form G, we set

G^{♭} := G|(V U − UV).

In terms of the q-expansion

G (Tate(q), ζ_{N}, ω_{can}) = ∑_{n=1}^{∞} b_{n}q^{n}

of G, the operators U and V satisfy

(3.8.3)  (G|U) (Tate(q), ζ_{N}, ω_{can}) = ∑_{n=1}^{∞} b_{np}q^{n},  (G|V) (Tate(q), ζ_{N}, ω_{can}) = ∑_{n=1}^{∞} b_{n}q^{np},

so that the q-expansion of G^{♭} is given by

(3.8.4)  G^{♭}(Tate(q), ζ_{N}, ω_{can}) = ∑_{(p,n)=1} b_{n}q^{n}.

Lemma 3.23. Let K be a quadratic imaginary field in which the prime (p) = 𝔭𝔭̄ splits, and let (A′, t′) be a point in C^{ord} corresponding to an elliptic curve A′ with complex multiplication by (an order in) K. Let G be a locally analytic p-adic modular form of weight k satisfying

T_{p}G = b_{p}G, ⟨p⟩G = ϵ_{G}(p)G.

Then

G^{♭}(A′, t′, ω′) = G(A′, t′, ω′) − \frac{ϵ_{G}(p)b_{p}}{p^{k}} G(𝔭 ∗ (A′, t′, ω′)) + \frac{ϵ_{G}(p)}{p^{k+1}}G(𝔭^{2} ∗ (A′, t′, ω′)),

where the action of ideals on CM triples is the one given in (1.4.8).

Proof. Because A′ has complex multiplication, its canonical subgroup is identified with the kernel A′[𝔭] of multiplication by 𝔭, and therefore,

V (A′, t′, ω′) = 𝔭 ∗ (A′, p^{−1}t′, pω′), [p]V^{2}(A′, t′, ω′) = 𝔭^{2} ∗ (A′, p^{−1}t′, pω′).

Therefore,

G^{♭}(A′, t′, ω′) = G((1 − T_{p}V + \frac{1}{p}[p]V^{2})(A′, t′, ω′))
= G(A′, t′, ω′) − b_{p}G(𝔭 ∗ (A′, p^{−1}t′, pω′)) + \frac{1}{p}G(𝔭^{2} ∗ (A′, p^{−1}t′, pω′))
= G(A′, t′, ω′) − \frac{ϵ_{G}(p)b_{p}}{p^{k}} G(𝔭 ∗ (A′, t′, ω′)) + \frac{ϵ_{G}(p)}{p^{k+1}}G(𝔭^{2} ∗ (A′, t′, ω′)).

The result follows. □

Proposition (3.24) below gives an explicit formula for G_{j}^{♭} in terms of the Atkin-Serre operator θ defined in equation (1.3.2) acting on the modular form f. Note that, for any j ≥ 0, the expression

θ^{−1−j}f^{♭} := lim_{t→−1−j} θ^{t}f^{♭}

is a p-adic modular form of weight r − 2j. (Cf. Théorème 5 (b) of [Se].)

Proposition 3.24. For all (E, t) ∈ C^{ord},

(3.8.5)  G_{j}^{♭}(E, t, ω) = j!θ^{−1−j}f^{♭}(E, t, ω).

(In particular the Coleman primitive F_{f}^{♭} of ω_{f^{♭}} is a rigid analytic section of L_{r}^{rig} over C^{ord}.)

Proof. For 0 ≤ j ≤ r, set F^{♭} := F_{f}^{♭} = F_{f}|(V U − UV). Then

G_{j}^{♭}(E, t, ω) = ⟨F^{♭}(E, t), ω^{j}η^{r−j}⟩.

Equation (3.8.5) amounts to the statement that

(3.8.6)  θG_{0}^{♭} = f^{♭},  θG_{j}^{♭} = jG_{j−1}^{♭}, for 1 ≤ j ≤ r.

We verify that this holds on q-expansions, working with the basis (ω_{can}, ξ_{can}) for the de Rham cohomology of the Tate curve which is described in equation (1.1.6) of Section 1.1. To check (3.8.6), note that

∇G_{0}^{♭}(Tate(q), ζ_{N}) = ∇ (G_{0}^{♭}(Tate(q), ζ_{N}, ω_{can})ω_{can}^{r})
= ∇ (⟨F^{♭}(Tate(q), ζ_{N}), ξ_{can}^{r}⟩ω_{can}^{r})
= ⟨ω_{f^{♭}}(Tate(q), ζ_{N}), ξ_{can}^{r}⟩ω_{can}^{r} + r⟨F^{♭}(Tate(q), ζ_{N}), ξ_{can}^{r}⟩ω_{can}^{r−1}ξ_{can} \frac{dq}{q}
= f^{♭}(Tate(q), ζ_{N}, ω_{can})ω_{can}^{r} \frac{dq}{q} + r⟨F^{♭}(Tate(q), ζ_{N}), ξ_{can}^{r}⟩ω_{can}^{r−1}ξ_{can} \frac{dq}{q} ,

where the last equality follows from (1.1.10).

After applying the inverse of the Kodaira-Spencer isomorphism and using (1.1.10) again, we find

˜∇G_{0}^{♭}(Tate(q), ζ_{N}) = f^{♭}(Tate(q), ζ_{N}, ω_{can})ω_{can}^{r+2} + r⟨F^{♭}(Tate(q), ζ_{N}), ξ_{can}^{r}⟩ω_{can}^{r+1}ξ_{can}.

Applying the unit root splitting Ψ_{Frob} to this identity then gives

Θ_{Frob}G_{0}^{♭}(Tate(q), ζ_{N}) = f^{♭}(Tate(q), ζ_{N}).

This proves (3.8.6) for j = 0, in light of Lemma 1.7. For the case j ≥ 1, we note that, because ⟨ω_{f^{♭}}, ω_{can}^{j}ξ_{can}^{r−j}⟩ = 0,

∇G_{j}^{♭}(Tate(q), ζ_{N}) = ∇ (G_{j}^{♭}(Tate(q), ζ_{N}, ω_{can})ω_{can}^{r−2j})
= ∇(⟨F^{♭}(Tate(q), ζ_{N}), ω_{can}^{j}ξ_{can}^{r−j}⟩ω_{can}^{r−2j})
= j⟨F^{♭}(Tate(q), ζ_{N})), ω_{can}^{j−1}ξ_{can}^{r−j+1}⟩ω_{can}^{r−2j} \frac{dq}{q}

+(r − 2j)⟨F^{♭}(Tate(q), ζ_{N})), ω_{can}^{j}ξ_{can}^{r−j}⟩ω_{can}^{r−2j−1}ξ_{can} \frac{dq}{q}

= jG_{j−1}^{♭}(Tate(q), ζ_{N}, ω_{can})ω_{can}^{r−2j} \frac{dq}{q}

+(r − 2j)G_{j}^{♭}(Tate(q), ζ_{N}, ω_{can})ω_{can}^{r−2j−1}ξ_{can} \frac{dq}{q} .

Applying σ^{−1} followed by the unit root splitting to this identity gives

Ψ_{Frob} ˜∇G_{j}^{♭}(Tate(q), ζ_{N}) = jG_{j−1}^{♭}(Tate(q), ζ_{N}, ω_{can})ω_{can}^{r+2−2j}.

Therefore,

Θ_{Frob}G_{j}^{♭}(Tate(q), ζ_{N}, ω_{can}) = jG_{j−1}^{♭}(Tate(q), ζ_{N}, ω_{can}),

and (3.8.6) follows from Lemma 1.7 for all 1 ≤ j ≤ r. This completes the proof of Proposition 3.24. (See also Lemma 9.2 of [Col3], where a similar result is proved.) □

4. Period integrals and central values of Rankin-Selberg L-functions

4.1. Rankin L-series and their special values. Let f = ∑ a_{n}e^{2πinz} ∈ S_{k}(Γ_{0}(N), ε_{f}) be a normalised newform. Write

L(f, s) = ∑_{n≥1} a_{n}n^{−s} = ∏_{q}(1 − α_{q}q^{−s})^{−1}(1 − β_{q}q^{−s})^{−1}

for its Hecke L-series, where the product on the right, taken over all the rational primes, should be taken as the definition of the parameters {α_{q}, β_{q}}. In particular, α_{q}β_{q} = q^{k−1}ε_{f}(q) if q does not divide N, and α_{q}β_{q} = 0 otherwise. Let N_{ε_{f}} denote the conductor of ε_{f}.

In this section, it will also be convenient to view f as a function on pairs (L, t), where L is a lattice in C and t is an element of exact order N in C/L. The lattice function f is determined by the rules

(4.1.1)  f(⟨1, τ⟩, 1/N) = f(τ),  for all τ ∈ H,
(4.1.2)  f(λL, λt) = λ^{−k}f(L, t),  for all λ ∈ C^{×},
(4.1.3)  f(L, at) = ε_{f}(a)f(L, t),  for all a ∈ (Z/NZ)^{×}.

Let w_{f} ∈ C^{×} be the scalar of norm one defined by the rule

(4.1.4)  w_{N}(f) = w_{f}f_{ρ},

where f_{ρ} ∈ S_{k}(Γ_{0}(N), ε̄_{f}) is the modular form obtained by applying complex conjugation to the coefficients of f and w_{N} is the Atkin-Lehner involution (which is described precisely in Lemma 5.2 and the discussion preceding it). We note that the Hecke L-series L(f, s) satisfies the functional equation

Λ(f, s) = w_{f}Λ(f_{ρ}, k − s),

where Λ(f, s) = (2π)^{−s}Γ(s)N^{s/2}L(f, s).

Let K be an imaginary quadratic field with discriminant −d_{K}, equipped with a fixed complex embedding. Recall that for any pair of integers (ℓ_{1}, ℓ_{2}), a Hecke character of K of infinity type (ℓ_{1}, ℓ_{2}) is a continuous homomorphism

χ : 𝔸_{K}^{×}⟶C^{×}

satisfying

χ(α · x · z_{∞}) = χ(x) · z_{∞}^{−ℓ_{1}}z̄_{∞}^{−ℓ_{2}}, for all α ∈ K^{×}, z_{∞} ∈ K_{∞}^{×}.

For each prime 𝔮 of K, let χ_{𝔮} : K_{𝔮}^{×}⟶C^{×} denote the local character associated to χ. The conductor of χ is the largest integral ideal 𝔣_{χ} of K such that χ_{𝔮}(u) = 1 for all u ∈ (1 + 𝔣_{χ}O_{K,𝔮})^{×} ↪ K_{𝔮}^{×}. In the usual way, we can identify χ with a character on O_{K}-ideals prime to 𝔣_{χ} by defining

(4.1.5)  χ(𝔞) = ∏_{𝔮|𝔞} χ_{𝔮}(π_{𝔮})^{v_{𝔮}(𝔞)},

where π_{𝔮} is any uniformizer at 𝔮, this assignment being independent of the choice of π_{𝔮}. As a function on ideals, χ satisfies χ((α)) = α^{ℓ_{1}}ᾱ^{ℓ_{2}} for all principal ideals (α) with α ≡ 1 mod 𝔣_{χ}.

The focus of this section is on the special values of the Rankin-Selberg L-function L(f × θ_{χ}, s) where θ_{χ} denotes the theta function associated to χ. For simplicity we will denote this L-function by L(f, χ, s). If we set α_{p^{j}} := α_{p}^{j} and β_{p^{j}} := β_{p}^{j}, then it can be defined as an Euler product of terms L_{𝔭}(f, χ, s) where for good 𝔭, i.e. for 𝔭 ∤ 𝔣_{χ}N,

L_{𝔭}(f, χ, s) = (1 − χ(𝔭)α_{N𝔭}(N𝔭)^{−s})^{−1}(1 − χ(𝔭)β_{N𝔭}(N𝔭)^{−s})^{−1}.

The local factors at ramified places are described in [Jac] §15. Indeed, up to a shift L(f, χ, s) is identified with the Rankin-Selberg L-function L(π_{f} ×π_{χ}, s), where π_{f} and π_{χ} are the automorphic representations of GL_{2}(𝔸_{Q}) associated to f and θ_{χ} respectively. More precisely, after normalizing π_{f} and π_{χ} to be unitary, we have

L(f, χ, s) = L (π_{f} × π_{χ}, s − \frac{k − 1 + ℓ_{1} + ℓ_{2}}{2} ) .

Set ℓ := |ℓ_{1} − ℓ_{2}| and ℓ_{0} := min(ℓ_{1}, ℓ_{2}). Define

L_{∞}(f, χ, s) = Γ_{C}(s − ℓ_{0})Γ_{C}(s − min(k − 1, ℓ) − ℓ_{0}),

where Γ_{C}(s) = 2 · (2π)^{−s}Γ(s), and set

Λ(f, χ, s) := L_{∞}(f, χ, s) · L(f, χ, s).

The function Λ(f, χ, s) (defined a priori in some right half plane) extends to a meromorphic function on C and satisfies a functional equation of the form

Λ(f, χ, s) = ϵ(f, χ, s)Λ(f_{ρ}, χ̄, k + ℓ_{1} + ℓ_{2} − s),

where f_{ρ} is as in (4.1.4) and ϵ(f, χ, s) is an epsilon factor again described in [Jac] §15. In the case of interest to us below, π_{f} × π_{χ} will be self-dual and the value of ϵ(f, χ, s) at the center of the critical strip, denoted ϵ(f, χ), is equal to ±1. If ε_{K} is the quadratic character associated to K and ε_{χ} is the Dirichlet character attached to χ by

ε_{χ} := χ|_{𝔸_{Q}^{×}} · N^{−(ℓ_{1}+ℓ_{2})},

then the function Λ(f, χ, s) is known to be holomorphic when ε_{f}ε_{χ}ε_{K} is non-trivial. (For more details on the above, see [Jac], § 19.)

An integer n is said to be critical (in the sense of Deligne) for L(f, χ, s) if none of the Gamma factors that occur on either side of the functional equation for L(f, χ, s) have a pole at s = n. The corresponding values of L(f, χ, s) will be called critical values. Deligne has made precise conjectures (proved by Shimura [Shim2]) that predict the rationality of these critical L-values over specific number fields, after dividing them by appropriate (ostensibly transcendental) periods. It turns out that the nature of the period depends qualitatively on the infinity type of χ. Indeed, assuming for the moment that χ is of type (0, ℓ) with ℓ ≥ 0, the form of the gamma factor L_{∞}(f, χ, s) shows that the following two cases arise naturally:

Case 1: ℓ ≤ k − 2. In this case the critical integers j for L(f, χ, s) are those in the closed segment [ℓ + 1, k − 1]. The transcendental part of L(f, χ, j) depends only on f and not on χ, and is expressible in terms of the Petersson inner product ⟨f, f⟩.

Case 2: ℓ ≥ k. In this case the critical integers j for L(f, χ, s) are those in the closed segment [k, ℓ]. The transcendental part of L(f, χ, j) depends only on K and not on f, and is expressible as a power of a CM period attached to K. (This period will be defined precisely in Section 5.1.)

We now return to considering characters χ of more general infinity type (ℓ_{1}, ℓ_{2}). It will be convenient in what follows to work with the L-function L(f, χ^{−1}, s). Note that the critical values of L(f, χ^{−1}, s) (as χ and s both vary) are completely captured by the critical values of L(f, χ^{−1}, 0) (as only χ is made to vary). This motivates the following definition.

Definition 4.1. A Hecke character χ of infinity type (ℓ_{1}, ℓ_{2}) is said to be critical if s = 0 is a critical point for L(f, χ^{−1}, s).

Let us define χ_{0} by χ_{0} := χ^{−1} · N^{ℓ_{1}} so that the infinity type of χ_{0} is (0, ℓ_{1} − ℓ_{2}). Then

L(f, χ^{−1}, s) = L(f, χ_{0}N^{−ℓ_{1}}, s) = L(f, χ_{0}, s + ℓ_{1}).

By the previous discussion applied to χ_{0} (and to χ_{0}^{ρ}: see remark below), the character χ of weight (ℓ_{1}, ℓ_{2}) is then critical if one of the following hypotheses is satisfied:

Case 1: 1 ≤ ℓ_{1}, ℓ_{2} ≤ k − 1. This implies that ℓ ≤ k − 2.

Case 2: ℓ_{1} ≥ k and ℓ_{2} ≤ 0, and Case 2′: ℓ_{1} ≤ 0 and ℓ_{2} ≥ k. In both these cases, ℓ ≥ k.

Let Σ^{(1)}, Σ^{(2)} and Σ^{(2′)} denote the set of Hecke characters satisfying the conditions in Case 1, Case 2 and Case 2′ respectively, so that the set Σ of all critical characters is the disjoint union

Σ = Σ^{(1)} ⊔ Σ^{(2)} ⊔ Σ^{(2′)}.

Remark 4.2. The weights of characters in Σ^{(1)} are the integer lattice points in the lightly shaded square in Figure 1, and those attached to characters in Σ^{(2)} are the lattice points in the darker lower right quadrant of this figure. The region Σ^{(2′)} is the reflection of Σ^{(2)} around the principal diagonal, and the map χ ↦ χ^{ρ} (where χ^{ρ} is the composition of χ with complex conjugation on 𝔸_{K}^{×}) interchanges these two regions.

A character χ ∈ Σ is said to be central critical if

ℓ_{1} + ℓ_{2} = k, ε_{χ} = ε_{f}.

The terminology is justified by the fact that in this case π_{f} ×π_{χ^{−1}} is self-dual and 0 is the central (critical) point for L(f, χ^{−1}, s). Let Σ_{cc} denote the set of central critical characters, and write (for i = 1, 2, 2′)

Σ_{cc}^{(i)} := Σ_{cc} ∩ Σ^{(i)}.

The weights of central critical characters are the lattice points on the central critical line which is depicted in Figure 1.

Figure 1. Critical and central critical weights for χ ↦ L(f, χ^{−1}, 0)

Remark 4.3. This article is concerned with the p-adic L-function obtained by interpolating the L-values L(f, χ^{−1}, 0) for χ in Σ^{(2)} or Σ^{(2′)}. Since this L-value is unchanged if χ is replaced by χ^{ρ}, we may assume that ℓ_{1} ≥ 0 and work simply with the region Σ^{(2)}. The main result of this paper (Theorem 5.13) relates the special values of this p-adic L-function at characters χ in Σ_{cc}^{(1)} (which is outside the range of interpolation) to the p-adic Abel-Jacobi images of generalised Heegner cycles. It would also be very interesting to study the values of this p-adic L-function at χ in Σ_{cc}^{(2′)}. We do not address this issue here. However, one could speculate that a study of the triple product L-function analogous to the one for the Rankin-Selberg Lfunction in this article may shed light on this issue. This intuition is suggested by the way in which the results of the present article are used in [BDP-ch] to yield information about the Katz p-adic L-function at critical characters that are outside the range of p-adic interpolation.

We assume henceforth that K satisfies the Heegner hypothesis for f i.e., that all the primes q | N are either split or ramified in K, and further that if q^{2} | N, then q is split in K. This implies that there exists a cyclic O_{K}-ideal 𝔑 of norm N. We fix once and for all such a choice of 𝔑. We also fix an integer c prime to Nd_{K}, and set (as in Sec. 1.4) 𝔑_{c} := 𝔑 ∩ O_{c}. Thus 𝔑_{c} is a proper O_{c}-ideal and O_{c}/𝔑_{c} ≃ O_{K}/𝔑 ≃ Z/NZ. Let U_{c} = Ô_{c}^{×} denote the corresponding compact open subgroup of 𝔸_{K,f}^{×}, so that U_{c} = ∏_{q} U_{c,q} with U_{c,q} := (O_{c} ⊗ Z_{q})^{×}. For ε any character of conductor N_{ε}|N, we define 𝔑_{ε} to be the unique ideal in O_{K} that divides 𝔑 and has norm equal to N_{ε}. Let ψ_{ε} be the composite homomorphism

(4.1.6)  U_{c} = Ô_{c}^{×} ↪ Ô_{K}^{×} → ∏_{𝔮|𝔑_{ε}}(O_{K,𝔮}/𝔑_{ε}O_{K,𝔮})^{×} ≃ ∏_{q|N_{ε}}(Z_{q}/N_{ε}Z_{q})^{×} \underrightarrow{∏ ε_{q}} C^{×}.

Equivalently, if we set 𝔑_{c,ε} := 𝔑_{ε} ∩ O_{c}, then ψ_{ε} is the composite

U_{c} = Ô_{c}^{×} → (Ô_{c}/𝔑_{c,ε}Ô_{c})^{×} ≃ (O_{K}/𝔑_{ε}O_{K})^{×} ≃ (Z/N_{ε}Z)^{×} ⟶^{ε^{−1}} C^{×}.

The following definition will be key in what follows.

Definition 4.4. A Hecke character χ of K is said to be of finite type (c, 𝔑, ε) if c divides 𝔣_{χ} and

χ|_{U_{c}} = ψ_{ε}.

Note that a character χ of finite type (c, 𝔑, ε) is necessarily unramified outside c𝔑_{ε}. Further, we may think of χ as a character on proper O_{c}-ideals prime to 𝔑_{c,ε}. Indeed, any such ideal 𝔞 is locally principal, i.e. 𝔞 = xO_{c} for some x = (x_{𝔮}) ∈ 𝔸_{K,fin}^{×}, and we set

(4.1.7)  χ(𝔞) := ∏_{𝔮∤𝔑_{ε}} χ_{𝔮}(x_{𝔮}).

This is independent of the choice of x since χ|_{O_{c,q}^{×}} = ψ_{ε}|_{O_{c,q}^{×}} = 1 for q ∤ N, and χ is unramified at the primes of K dividing N but not dividing 𝔑_{ε}. Viewed in this manner, χ satisfies

(4.1.8)  χ((α)) = α^{ℓ_{1}}ᾱ^{ℓ_{2}}ε(α mod 𝔑_{ε})

for any α ∈ K^{×} that is a unit at all the primes dividing 𝔑_{ε}.

Let Σ_{cc}(𝔑) denote the set of those characters in Σ_{cc}^{(1)} ⊔ Σ_{cc}^{(2)} that are of finite type (c, 𝔑, ε_{f}) and that satisfy the following auxiliary condition: the local sign ε_{q}(f, χ^{−1}) = +1 for all finite primes q. In view of our other hypotheses, this condition is automatic except possibly at those primes q lying in the set

S(f) := {q : q | (N, d_{K}), q ∤ N_{ε_{f}} }.

For i = 1, 2, we define Σ_{cc}^{(i)}(𝔑) by

Σ_{cc}^{(i)}(𝔑) := Σ_{cc}^{(i)} ∩ Σ_{cc}(𝔑),

so that Σ_{cc}(𝔑) is the disjoint union:

Σ_{cc}(𝔑) = Σ_{cc}^{(1)}(𝔑) ⊔ Σ_{cc}^{(2)}(𝔑).

For χ ∈ Σ_{cc}(𝔑), writing (k + j, −j) for the weight of χ, we see that χ ∈ Σ_{cc}^{(2)}(𝔑) or Σ_{cc}^{(1)}(𝔑) according as j ≥ 0 or j ∈ [−(k − 1), −1]. Let χ ∈ Σ_{cc}^{(2)}(𝔑) be a Hecke character of infinity type (k + j, −j). Recall the Shimura-Maass operator δ_{k} of equation (1.2.9) and let

δ_{k}^{j} = δ_{k+2j−2} ⋯ δ_{k+2}δ_{k}

be the differential operator sending holomorphic modular forms of weight k to nearly holomorphic modular forms of weight k + 2j. The modular form δ_{k}^{j}f can also be viewed as a function on pairs (L, t) consisting of a lattice L in C and an element t of order N in C/L, satisfying the homogeneity properties of (4.1.3) with k replaced by k + 2j.

In what follows, we will also fix a generator t of 𝔑_{c}^{−1}/O_{c} ≃ Z/NZ. Let 𝔞 be a proper O_{c}-ideal prime to 𝔑_{c} and choose α ∈ K^{×} such that 𝔟 := α𝔞 ⊂ O_{c} and α ≡ 1 mod 𝔑. Then the image of t under the composite map

𝔑_{c}^{−1}/O_{c} → 𝔑_{c}^{−1}𝔟^{−1}/𝔟^{−1} ⟶^{·α} 𝔑_{c}^{−1}𝔞^{−1}/𝔞^{−1}

is independent of the choice of α, and will be denoted t_{𝔞}. Thus the choice of t gives rise to a generator t_{𝔞} of 𝔑_{c}^{−1}𝔞^{−1}/𝔞^{−1} for every proper O_{c}-ideal 𝔞 prime to 𝔑_{c}.

Lemma 4.5. Let 𝔞 be any proper O_{c}-ideal prime to 𝔑_{c} and suppose χ is a Hecke character in Σ_{cc}^{(2)}(𝔑) of infinity type (k + j, −j). With t fixed, the expression

(4.1.9)  χ^{−1}(𝔞)N𝔞^{−j} · δ_{k}^{j}f(𝔞^{−1}, t_{𝔞})

depends only on the class of 𝔞 in Pic(O_{c}).

Proof. Note that since 𝔞 is prime to 𝔑_{c}, it is certainly prime to 𝔑_{c,ε} as well and so the expression χ^{−1}(𝔞) is well defined. The lemma then follows immediately from the equations (4.1.2) (with f replaced by δ_{k}^{j}(f) and k by k + 2j), (4.1.3) and (4.1.8). □

Theorem 4.6. Let f be a normalised eigenform in S_{k}(Γ_{0}(N), ε_{f}) and let χ ∈ Σ_{cc}^{(2)}(𝔑) be a Hecke character of K of infinity type (k + j, −j). Suppose also that c and d_{K} are odd, and let w_{K} denote the number of roots of unity in K. Then

C(f, χ, c) · L(f, χ^{−1}, 0) = | ∑_{[𝔞]∈Pic(O_{c})} χ^{−1}(𝔞)N𝔞^{−j} · (δ_{k}^{j}f)(𝔞^{−1}, t_{𝔞})|^{2} ,

where the representatives 𝔞 of the ideal classes in Pic(O_{c}) are chosen to be prime to 𝔑_{c} and the constant C(f, χ, c) is given by

C(f, χ, c) = \frac{1}{4}π^{k+2j−1}Γ(j + 1)Γ(k + j)w_{K}|d_{K}|^{1/2} · c vol(O_{c})^{−ℓ} · 2^{#S_{f}} · ∏_{q|c} \frac{(q − ε_{K}(q))}{q − 1}.

Remark 4.7. The restriction that c and d_{K} are odd is made for convenience to simplify the local calculations in Section 4.6 at primes dividing cd_{K}.

The rest of this chapter will be devoted to proving Theorem 4.6 using Waldspurger’s results relating period integrals to L-values. The reader whose main interest is in p-adic methods can take this result on faith and continue reading from Sec. 5.1 onwards.

4.2. Differential operators. We recall some general facts about the Shimura-Maass operators that were introduced in Sec. 1.2 and appear in the statement of the theorem above. Let Γ be a congruence subgroup of SL_{2}(Z) and denote by C_{k}^{∞}(Γ) the space of C^{∞}-modular forms of weight k on Γ. We also denote by C̃_{k}^{∞}(Γ) the space of C^{∞}-functions on H such that

f(γz) = (c′z + d′)^{k}|c′z + d′|^{−k}f(z)

for all γ = ( \begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix} ) ∈ Γ. (For the moment we will use the symbol f for an arbitrary modular form in

C_{k}^{∞}(Γ) or C̃_{k}^{∞}(Γ).) Recall that the weight k Shimura-Maass raising operator δ_{k} : C_{k}^{∞}(Γ) → C_{k+2}^{∞}(Γ) is defined by

(4.2.1)  δ_{k}(f) = \frac{1}{2πi} (\frac{∂}{∂z} + \frac{k}{z − z̄}) f.

Via the isomorphism

(4.2.2)  C_{k}^{∞}(Γ) ≃ C̃_{k}^{∞}(Γ),  f(z) ↦ f̃(z) := f(z)y^{k/2},

we see that δ_{k} is identified with −\frac{1}{4π} R_{k} where

(4.2.3)  R_{k} : C̃_{k}^{∞}(Γ) → C̃_{k+2}^{∞}(Γ),  R_{k}(f) = ((z − z̄)\frac{∂}{∂z} + \frac{k}{2} ) f.

Let us define (following the discussion in [Bump] §2.1)

(4.2.4)  L_{k} : C̃_{k}^{∞}(Γ) → C̃_{k−2}^{∞}(Γ),  L_{k}(f) = − ((z − z̄)\frac{∂}{∂z̄} + \frac{k}{2} ) f,

and

(4.2.5)  Δ_{k} : C̃_{k}^{∞}(Γ) → C̃_{k}^{∞}(Γ),  Δ_{k}(f) = −y^{2} (\frac{∂^{2}}{∂x^{2}} + \frac{∂^{2}}{∂y^{2}} ) + iky \frac{∂}{∂x}.

These operators satisfy

(4.2.6)  Δ_{k} = −L_{k+2}R_{k} − \frac{k}{2} (1 + \frac{k}{2} ) = −R_{k−2}L_{k} + \frac{k}{2} (1 − \frac{k}{2} ) .

Note that via the isomorphism (4.2.2), the lowering operator L_{k} corresponds to f ↦ 2i\frac{∂}{∂z̄}f on C_{k}^{∞}(Γ). Thus if f is holomorphic, then L_{k}(f̃) = 0.

Definition 4.8. Let j be a nonnegative integer and f ∈ C̃_{k}^{∞}(Γ). Then R^{j}f is defined by

R^{j}f = (R_{k+2j−2} ∘ R_{k+2j−4} ∘ ⋯ ∘ R_{k+2} ∘ R_{k})f.

Lemma 4.9. Suppose that f ∈ C_{k}^{∞}(Γ) is holomorphic. Then for j ≥ 0, the form R^{j}f̃ is an eigenfunction of Δ_{k+2j} with eigenvalue μ_{j} + λ_{j} where μ_{j} := j(k + j − 1) and λ_{j} := \frac{k+2j}{2} (1 − \frac{k+2j}{2} ).

Proof. Since f is holomorphic, we have L_{k}(f̃) = 0. Hence Δ_{k}f̃ = \frac{k}{2} (1 − \frac{k}{2} ) by (4.2.6) and the result holds for j = 0. We now work inductively, assuming the result holds for j − 1. By (4.2.6) again,

Δ_{k+2j}R^{j}f̃ = (−R_{k+2j−2}L_{k+2j} + λ_{j}) R^{j}f̃
= −R_{k+2j−2}L_{k+2j}R_{k+2j−2}R^{j−1}f̃+ λ_{j}R^{j}f̃
= R_{k+2j−2} (Δ_{k+2j−2} + \frac{k + 2j − 2}{2} (1 + \frac{k + 2j − 2}{2})) R^{j−1}f̃+ λ_{j}R^{j}f̃
= R_{k+2j−2} (μ_{j−1} + λ_{j−1} + \frac{k + 2j − 2}{2} (1 + \frac{k + 2j − 2}{2})) R^{j−1}f̃+ λ_{j}R^{j}f̃
= R_{k+2j−2} (μ_{j−1} + k + 2j − 2) R^{j−1}f̃+ λ_{j}R^{j}f̃
= (μ_{j−1} + k + 2j − 2 + λ_{j}) R^{j}f̃ = (μ_{j} + λ_{j})R^{j}f̃. □

Definition 4.10. Let f, g ∈ C_{k}^{∞}(Γ) and suppose at least one of f or g is a cusp form. Then set

⟨f, g⟩ = \frac{1}{[SL_{2}(Z) : Γ]} ∫_{Γ\H} f(z)\overline{g(z)}y^{k} \frac{dxdy}{y^{2}} .

Likewise, for f, g ∈ C̃_{k}^{∞}(Γ) with at least one being cuspidal, we set

⟨f, g⟩ = \frac{1}{[SL_{2}(Z) : Γ]} ∫_{Γ\H} f(z)\overline{g(z)}\frac{dxdy}{y^{2}} .

Clearly, for f, g ∈ C_{k}^{∞}(Γ), we have ⟨f, g⟩ = ⟨f̃, g̃⟩.

Lemma 4.11. Suppose that f, g ∈ C_{k}^{∞}(Γ) are holomorphic. Then

(4.2.7)  ⟨R^{j}f̃, R^{j}g̃⟩ = \frac{Γ(j + 1)Γ(k + j)}{Γ(k)}⟨f̃, g̃⟩,

and

(4.2.8)  ⟨δ_{k}^{j}f, δ_{k}^{j}g⟩ = \frac{1}{(4π)^{2j}} \frac{Γ(j + 1)Γ(k + j)}{Γ(k)}⟨f, g⟩.

Proof. Clearly (4.2.7) and (4.2.8) are equivalent. We will prove (4.2.7) inductively. Invoking [Bump] Prop. 2.1.3, equation (4.2.6) and Lemma 4.9 in turn, we find

⟨R^{j}f̃, R^{j}g̃⟩ = ⟨R^{j−1}f̃, −L_{k+2j}R_{k+2j−2}R^{j−1}g̃⟩
= ⟨R^{j−1}f̃, (Δ_{k+2j−2} + \frac{k + 2j − 2}{2} (1 + \frac{k + 2j − 2}{2})) R^{j−1}g̃⟩
= ⟨R^{j−1}f̃, (μ_{j−1} + λ_{j−1} + \frac{k + 2j − 2}{2} (1 + \frac{k + 2j − 2}{2})) R^{j−1}g̃⟩ = μ_{j}⟨R^{j−1}f̃, R^{j−1}g̃⟩.

Hence

⟨R^{j}f̃, R^{j}g̃⟩ = ⟨f̃, g̃⟩ · ∏_{1≤t≤j} μ_{t} = \frac{Γ(j + 1)Γ(k + j)}{Γ(k)}⟨f̃, g̃⟩. □

4.3. Period integrals and values at CM points. Let A_{0} := C/O_{c} and t_{0} be the 𝔑-torsion point on A_{0} corresponding to our choice of t ∈ 𝔑_{c}^{−1}/O_{c}. The pair (A_{0}, t_{0}) determines a point P_{A_{0}} on the modular curve X_{1}(N). Let τ ∈ H be any any point lying over P_{A_{0}}. Thus there is a unique isomorphism

A_{τ} := C/Zτ + Z ≃^{·Λ_{τ}} C/O_{c}

sending [1/N] to t_{0}, which on tangent spaces is given by multiplication by a scalar Λ_{τ} ∈ K^{×}. Hence O_{c} = Λ_{τ} (Zτ + Z) and

\frac{Λ_{τ}}{N} ≡ t mod O_{c}.

Thus

(4.3.1)  Λ_{τ} ∈ \overline{𝔑_{c}},  and  (Λ_{τ} , 𝔑_{c}) = 1.

Let ξ : K ↪ M_{2}(Q) be the embedding that describes the action of K on H_{1}(A_{τ} (C), Q) with respect to the basis (τ, 1) i.e. given by

α · [ \begin{matrix}τ \\ 1 \end{matrix} ] = ξ(α) [ \begin{matrix}τ \\ 1 \end{matrix} ] .

Explicitly, for a, b ∈ Q,

(4.3.2)  ξ(a + bτ) = ( \begin{matrix}a + b Tr(τ) & −bNτ \\ b & a \end{matrix} ) .

Let M_{0}(N) be the order defined by

M_{0}(N) := \{( \begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix} ) ∈ M_{2}(Z) : c′ ≡ 0 mod N\} .

Then, via the embedding ξ,

K ∩ M_{0}(N) = End(A_{τ} , ⟨[1/N]⟩) = End(C/O_{c}, ⟨t⟩) = O_{c},

so that ξ is a Heegner embedding of conductor c. A different choice of τ will give an embedding ξ′ that is conjugate to ξ by an element of Γ_{0}(N). Note that ξ gives rise to a map of algebraic groups

ξ : Res_{K/Q}𝔾_{m} ↪ GL_{2,Q}

and hence a map on adelic points ξ_{𝔸} : 𝔸_{K}^{×} ↪ GL_{2}(𝔸_{Q}). We consider 𝔸_{K}^{×} as a subgroup of GL_{2}(𝔸_{Q}) via this embedding.

As in the previous section, let δ_{k}^{j}f denote the nearly holomorphic modular form of weight ℓ := k + 2j obtained by applying the Shimura-Maass differential operator j times to f. We use the embedding ξ to associate to the classical modular form δ_{k}^{j}f an automorphic form F^{j} on GL_{2}(𝔸_{Q}) as follows. First, let

U′_{q} := (M_{0}(N) ⊗ Z_{q})^{×}, U′ := { \atop M_{0}(N)}^{×} = ∏_{q} U′_{q} ⊂ GL_{2}(𝔸_{f})

and define a character ω_{f} = ∏_{q} ω_{f,q} of U′ by setting

ω_{f,q} ( \begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix} ) = ε_{f,q}(d′)

for ( \begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix} ) ∈ U′_{q}. Now, for g ∈ GL_{2}(𝔸_{Q}), write

g = γ · (uγ_{∞}), with γ ∈ GL_{2}(Q), u ∈ U′, γ_{∞} ∈ GL_{2}(R)^{+}.

Then set

F^{j}(g) = δ_{k}^{j}(f)(γ_{∞}(τ))j(γ_{∞}, τ)^{−ℓ}ω_{f}(u),

where we define

J(γ′, z) := c′z + d′  and  j(γ′, z) := det(γ′)^{−1/2}(c′z + d′),

for any γ′ = ( \begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix} ) ∈ GL_{2}(R). One checks easily that this definition is independent of the choice of

decomposition of g. Further, for any α ∈ K_{∞}^{×},

F^{j}(gα) = F^{j}(g)j(α, τ)^{−ℓ} = α^{−ℓ}N_{K}(α)^{ℓ/2}F^{j}(g).

Here N_{K} = N ∘ N_{K/Q} is the usual norm character on K, N being the norm character on Q.

Lemma 4.12. The restriction of the character ω_{f} of U′ to U_{c} (via the embedding ξ_{𝔸}) is ψ_{ε_{f}}.

Proof. For q ∤ N, the restrictions of ω_{f} to U′_{q} and of ψ_{ε_{f}} to U_{c,q} are both trivial. Suppose therefore that q divides N. Let a + bτ ∈ O_{c} ∩ U_{c,q}. By (4.3.2), we have a ∈ Z and b ∈ NZ. Since N/Λ_{τ} lies in 𝔑_{c} ⊗ Z_{q} and Λ_{τ} τ ∈ O_{c}, the element Nτ = (N/Λ_{τ}) · Λ_{τ} τ also lies in 𝔑_{c} ⊗ Z_{q}, so that

ψ_{ε_{f} ,q}(a + bτ) = ε_{f,q}(a) = ω_{f,q}(ξ_{q}(a + bτ)).

Since O_{c} ∩ U_{c,q} is dense in U_{c,q}, it follows that ψ_{ε_{f}} (u) = ω_{f}(ξ_{𝔸}(u)) for all u ∈ U_{c,q} ⊆ U_{c}. □

Proposition 4.13. Suppose χ ∈ Σ_{cc}^{(2)}(𝔑) is of infinity type (k + j, −j). Let η and η′ be Grossencharacters defined by

η := χ^{−1}N_{K}^{−j}, η′ := ηN_{K}^{ℓ/2},

so that η′ is unitary. Then

\frac{1}{h_{c}} ∑_{[𝔞]∈Pic(O_{c})} χ^{−1}(𝔞)N𝔞^{−j} · (δ_{k}^{j}f)(𝔞^{−1}, t_{𝔞}) = (2πi)^{ℓ}Λ_{τ}^{−ℓ} ∫_{K^{×}K_{∞}^{×}\𝔸_{K}^{×}} F^{j}(ξ_{𝔸}(x)) · η′(x)d^{×}x,

where h_{c} := # Pic(O_{c}) and the measure d^{×}x on K^{×}K_{∞}^{×} \ 𝔸_{K}^{×} is chosen to have total volume 1.

Proof. Let us pick elements y_{i} ∈ Ô_{c} such that 𝔸_{K}^{×} = ⊔_{i=1}^{h}K^{×} · U_{c} · K_{∞}^{×} · y_{i}. We may assume that we have picked y_{i} to satisfy

(4.3.3)  y_{i,𝔮} ≡ 1 mod 𝔑O_{K,𝔮} for 𝔮 | 𝔑.

Let 𝔞_{i} := y_{i}O_{c} be the associated proper O_{c}-ideal, so that

(4.3.4)  η(y_{i}) = η(𝔞_{i}) = χ^{−1}(𝔞_{i})N𝔞_{i}^{−j}.

Let U′′ := ∏_{q} U′′_{q} be the subgroup of U′ defined by U′′_{q} := U′_{q} if q ∤ N and

U′′_{q} := \{( \begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix} ) ∈ U′_{q} : d′ ≡ 1 mod N\} .

By strong approximation for GL_{2}, we may write

ξ_{𝔸}(y_{i}) = g_{i}(g_{U,i} · γ_{i}) with g_{i} ∈ GL_{2,Q}, g_{U,i} ∈ U′′, γ_{i} ∈ GL_{2}(R)^{+}.

Since g_{i}γ_{i} = 1, we have γ_{i}^{−1} = g_{i} ∈ GL_{2}(Q)^{+}. Further, since ξ is a Heegner embedding, we have g_{i}g_{U,i} ∈ { \atop M_{0}(N)} and consequently γ_{i}^{−1} ∈ { \atop M_{0}(N)} ∩ GL_{2}(Q)^{+}. i.e.

(4.3.5)  γ_{i}^{−1} = ( \begin{matrix}a_{i} & b_{i} \\ c_{i} & d_{i} \end{matrix} ) ∈ M_{2}(Z) ∩ GL_{2}(Q)^{+}, c_{i} ∈ NZ.

In fact, on account of (4.3.3) and the fact that Nτ ∈ 𝔑_{c} ⊗ Z_{q} for q | N (see the proof of Lemma 4.12 above), we also have d_{i} ≡ 1 mod N. Now, for u ∈ U_{c},

F^{j}(ξ_{𝔸}(xu)) = F^{j}(ξ_{𝔸}(x))ω_{f}(ξ_{𝔸}(u)) = F^{j}(ξ_{𝔸}(x))ε_{f}(u).

Hence

∫_{K^{×}K_{∞}^{×}\𝔸_{K}^{×}} F^{j}(ξ_{𝔸}(x)) · η′(x)d^{×}x = \frac{1}{h_{c}} ∑_{i=1}^{h_{c}} δ_{k}^{j}(f)(γ_{i}τ)j(γ_{i}, τ)^{−ℓ}ω_{f}(g_{U,i})η′(y_{i})

= \frac{1}{h_{c}} ∑_{i=1}^{h_{c}} δ_{k}^{j}(f)(γ_{i}τ)J(γ_{i}, τ)^{−ℓ}η(y_{i}),

since ω_{f}(g_{U,i}) = 1. Taking into account (4.3.4), it will suffice to show that

(2πi)^{ℓ}Λ_{τ}^{−ℓ}δ_{k}^{j}(f)(γ_{i}τ)J(γ_{i}, τ)^{−ℓ} = (δ_{k}^{j}f)(𝔞_{i}^{−1}, t_{𝔞_{i}}).

From the choice of γ_{i}, we see that the class of γ_{i}τ in X_{1}(N) corresponds to the pair (C/𝔞_{i}^{−1}, t_{𝔞_{i}}), and there is a unique isomorphism

C/(Zγ_{i}τ + Z) ≃^{·λ_{i}} C/𝔞_{i}^{−1},

sending [1/N] to t_{𝔞_{i}}, with a scalar λ_{i} ∈ K^{×}. Note that

J(γ_{i}, τ)^{−1} = J(γ_{i}^{−1}, γ_{i}τ) = c′(γ_{i}τ) + d′.

The scalar λ_{i} may then be identified from the fact that there is a commutative diagram:

\xymatrix{C/(Zτ + Z) \ar[r]^{J(γ_{i},τ)^{−1}} \ar[d]^{Λ_{τ}} & C/(Zγ_{i}τ + Z) \ar[d]^{λ_{i}} \\ C/O_{c} \ar[r] & C/𝔞_{i}^{−1}.}

Thus λ_{i} = Λ_{τ} · J(γ_{i}, τ), and

δ_{k}^{j}(f)(𝔞_{i}^{−1}, t_{𝔞_{i}}) = δ_{k}^{j}(f)(C/(Zγ_{i}τ + Z), λ_{i}^{−1}dz, [1/N])
= Λ_{τ}^{−ℓ}(2πi)^{ℓ}δ_{k}^{j}(f)(γ_{i}τ)J(γ_{i}, τ)^{−ℓ}. □

In the next few sections we will study the period integral

(4.3.6)  L_{η′,ξ}(F^{j}) := ∫_{K^{×}K_{∞}^{×}\𝔸_{K}^{×}} F^{j}(ξ_{𝔸}(x)) · η′(x)d^{×}x

using the method of Waldspurger.

4.4. Explicit theta lifts. Let ψ denote the additive character of 𝔸/Q given by ψ((x_{v})_{v}) = ∏_{v} ψ_{v}(x_{v}), where

ψ_{∞}(x) = e^{2πix},  ψ_{q}(x) = e^{−2πix}  for x ∈ Z [\frac{1}{q}] ⊂ Q_{q}.

Let (V, ⟨, ⟩) be an even dimensional orthogonal space over Q, and denote by O(V) (resp. GO(V)) its isometry group (resp. orthogonal similitude group). Recall the Weil representation r_{ψ} = ∏_{v} r_{ψ,v} of the group SL_{2}(𝔸) × O(V)(𝔸) on the Schwartz space S(V (𝔸)). On the orthogonal group, r_{ψ,v} is given by

r_{ψ,v}(g)φ(x) = φ(g^{−1} · x) for g ∈ O(V)(Q_{v}), φ ∈ S(V (Q_{v})).

On SL_{2}(Q_{v}), the representation r_{ψ,v} is described by its action on the matrices

U(a) := ( \begin{matrix}1 & a \\ 0 & 1 \end{matrix} ) , D(a) := ( \begin{matrix}a & 0 \\ 0 & a^{−1} \end{matrix} ) , W := ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) ,

by the equations

r_{ψ,v}(U(a))φ(x) = ψ_{v}(\frac{1}{2}⟨ax, x⟩)φ(x),
r_{ψ,v}(D(a))φ(x) = χ_{V,v}(a)|a|_{v}^{dim(V)/2}φ(ax),
r_{ψ,v}(W)φ(x) = γ_{V,v}φ̂(x),

where χ_{V,v} is a quadratic character and γ_{V,v} is an eighth root of unity, that can be read off from [JL] §1. In the cases of interest to us, they can also be found listed in the table in [P], §3.4. The Fourier transform φ̂ is defined by

φ̂(x) = ∫_{V (Q_{v})} φ(y)ψ_{v}(⟨y, x⟩)dy,

the measure dy on V (Q_{v}) being chosen such that ˆ φ̂(x) = φ(−x).

We will need to extend the Weil representation to similitude groups, following Harris-Kudla [HK1]. Let R be the group defined by:

R := {(g, h) ∈ GL_{2} × GO(V) : det(g) = ν(h)}

where ν denotes the similitude character of GO(V). Then r_{ψ} can be extended to R(𝔸) by

r_{ψ}(g, h)φ = r_{ψ} (g · ( \begin{matrix}1 & 0 \\ 0 & det g^{−1} \end{matrix} )) L(h)φ,

where

L(h)φ(x) = |ν(h)|^{− dim(V)/4}φ(h^{−1}x).

Let GO(V)^{0} denote the algebraic connected component of GO(V). If F is an automorphic form on GL_{2}(𝔸) and φ ∈ S(V (𝔸)), we define for h ∈ GO(V)(𝔸),

θ_{φ}(F)(h) := ∫_{SL_{2}(Q)\SL_{2}(𝔸)} ∑_{x∈V (Q)} r_{ψ}(gg′, h)φ(x)F (gg′)d^{(1)}g,

where g′ is chosen such that det(g′) = ν(h). Likewise, in the opposite direction, if F′ is an automorphic form on GO(V)^{0}(𝔸), and g ∈ GL_{2}(𝔸) is such that det(g) ∈ ν(GO(V)(𝔸)), we set

θ_{φ}^{t}(F′)(g) := ∫_{O(V)(Q)\O(V)(𝔸)} ∑_{x∈V (Q)} r_{ψ}(g, hh′)φ(x)F′(hh′)dh,

where h′ ∈ GO(V)^{0}(𝔸) is chosen such that det(g) = ν(h′). (We refer the reader to [P], §1, for the choices of measures in the above and in what follows. ) If π (resp. Π) is an automorphic representation of GL_{2}(𝔸) (resp. of GO(V)^{0}(𝔸)), we define

θ(π) := {θ_{φ}(F) : F ∈ π, φ ∈ S(V (𝔸))};
θ^{t}(Π) := {θ_{φ}^{t}(F′) : F′ ∈ Π, φ ∈ S(V (𝔸))}.

Now set V := M_{2}(Q) and consider V as an orthogonal space over Q with bilinear form

⟨x, y⟩ = \frac{1}{2}(xy^{ι} + yx^{ι}), ( \begin{matrix}a & b \\ c & d \end{matrix} )^{ι} = ( \begin{matrix}d & −b \\ −c & a \end{matrix} ) .

The associated quadratic form is just x ↦ xx^{ι} = det(x). The group GO(V)^{0} is identified with the quotient Q^{×} \ GL_{2} × GL_{2} via the map (α, β) ↦ δ(α, β) where δ(α, β)(x) = αxβ^{−1}. Thus an automorphic representation of GO(V)^{0}(𝔸) is identified with a pair (π_{1}, π_{2}) of representations of GL_{2}(𝔸), such that the product of the central characters of π_{1} and π_{2} is trivial. To ease notation, we will often just write (α, β) to denote the element δ(α, β).

Let π denote the (unitary) automorphic representation of GL_{2}(𝔸) associated to f. The following theorem is the classical Jacquet-Langlands correspondence realized using theta functions, and is essentially due to Shimizu [SH]. (See also [Wa] §3.2.)

Theorem 4.14. (1) θ(π̄) = π̄ × π, where π̄ = π^{∨} = π ⊗ ε_{f}^{−1}.

(2) θ^{t}(π × π̄) = π.

We will need a statement involving specific forms in π and π̄ and explicit theta functions i.e. explicit choices of Schwartz functions. For any finite prime q, let q^{n_{q}} be the exact power of q dividing N and for any set A, let I_{A} denote the characteristic function of A. For q a prime dividing N, we will set

(4.4.1)  φ_{q}^{1} ( \begin{matrix}a & b \\ c & d \end{matrix} ) := \{ \begin{matrix}I_{Z_{q}}(a)I_{Z_{q}}(b)I_{q^{nq} Z_{q}}(c)I_{Z_{q}}(d), if q ∤ N_{ε_{f}} , \\ ε_{f,q}(d)I_{Z_{q}}(a)I_{Z_{q}}(b)I_{q^{nq} Z_{q}}(c)I_{Z_{q}^{×}}(d), if q | N_{ε_{f}}; \end{matrix}

(4.4.2)  φ_{q}^{2} ( \begin{matrix}a & b \\ c & d \end{matrix} ) := \{ \begin{matrix}\frac{1}{q}I_{Z_{q}}(a)I_{Z_{q}}(b)I_{q^{n−1}Z_{q}}(c)I_{Z_{q}}(d), if q ∤ N_{ε_{f}} , \\ \frac{1}{q}ε_{f,q}(d)I_{Z_{q}}(a)I_{Z_{q}}(b)I_{q^{nq −1}Z_{q}}(c)I_{Z_{q}^{×}}(d), if q | N_{ε_{f}} . \end{matrix}

Let Σ denote the set of primes dividing N. For now we will fix a subset Ξ of Σ and consider the following Schwartz function: φ^{Ξ} := ⊗_{q}φ_{q}^{Ξ} where

(i) For q ∤ N, φ_{q}^{Ξ} = I_{M_{0}(N)⊗Z_{q}} = I_{M_{2}(Z_{q})}; (ii) For q | N, φ_{q}^{Ξ} = φ_{q}^{1} or φ_{q}^{2} according as q ∉ Ξ or q ∈ Ξ; (iii) For q = ∞, we identify M_{2}(R) = (K ⊗ R) + (K ⊗ R)^{⊥} = C + C^{⊥} and set φ_{∞}^{Ξ} = φ_{∞}, with

(4.4.3)  φ_{∞}(u + v) = ū^{ℓ}p_{j}(4π⟨v, v⟩)e^{−2π(|⟨u,u⟩|+|⟨v,v⟩|)},

for u ∈ C, v ∈ C^{⊥}, where p_{j} denotes the jth Laguerre polynomial

p_{j}(X) = ∑_{s=0}^{j} \binom{j}{s}\frac{(−X)^{s}}{s!} .

Lemma 4.15. Suppose κ_{θ} := ( \begin{matrix}cos θ & − sin θ \\ sin θ & cos θ \end{matrix} ) ∈ SO_{2}(R) and κ_{1}, κ_{2} ∈ (K ⊗ R)^{(1)} ⊂ GL_{2}(R). Then

r_{ψ}(κ_{θ}, (κ_{1}, κ_{2}))φ_{∞} = e^{ikθ} · κ_{1}^{ℓ} · κ_{2}^{−ℓ}φ_{∞}.

Proof. This is [Xue], Prop. 2.2.5. □

For q | N, let us set U_{q}^{1} := U′_{q} (recall that U′_{q} was defined to be (M_{0}(N) ⊗ Z_{q})^{×}) and

U_{q}^{2} := \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(Z_{q}) : a, d ∈ Z_{q}^{×}, b ∈ qZ_{q}, c ∈ q^{n_{q}−1}Z_{q}\} .

We also set U_{q}^{Ξ} equal to U′_{q} if q ∤ N and equal to U_{q}^{1} or U_{q}^{2} according as q ∉ Ξ or q ∈ Ξ, if q | N.

Lemma 4.16. Let q be a finite prime and suppose α, β ∈ U′_{q}, γ ∈ U_{q}^{Ξ} are such that

det(α) = det(β) · det(γ)^{−1},

so that (α, (β, γ)) may be viewed as an element of R(Q_{q}).

(1) Suppose q ∤ N_{ε_{f}}. Then

r_{ψ}(α, (β, γ))φ_{q}^{Ξ} = φ_{q}^{Ξ}.
(2) Suppose q | N_{ε_{f}}. Then
r_{ψ}(α, (β, γ))φ_{q}^{Ξ} = ε_{f,q}(a(α))ε_{f,q}(d(β)^{−1}d(γ))φ_{q}^{Ξ},

where for any matrix α in GL_{2}, we define a(α) and d(α) to be the upper left and lower right entries of α respectively.

Proof. Let us write φ_{q} instead of φ_{q}^{Ξ} for simplicity. Clearly we may assume that

det(α) = det(β) det(γ)^{−1} = 1.

Then

r_{ψ}(α, (β, γ))φ_{q}(x) = r_{ψ}(α)L(β, γ)φ_{q}(x) = r_{ψ}(α)φ_{q}(β^{−1}xγ).

In case (1), we have φ_{q}(β^{−1}xγ) = φ_{q}(x), while in case (2), φ_{q}(β^{−1}xγ) = ε_{f,q}(d(β)^{−1}d(γ))φ_{q}(x). So it suffices to consider the action of r_{ψ}(α) on φ_{q}. Let us first check case (1). If further q ∤ N, then α is in the subgroup generated by matrices of the form D(a), U(y) and W with a ∈ Z_{q}^{×} and y ∈ Z_{q}. Thus we may assume that α is in fact one of these three possibilities. Since φ_{q} = I_{M_{2}(Z_{q})} in this case, one checks easily that

(4.4.4)  r_{ψ}(D(a))φ_{q}(x) = φ_{q}(ax) = φ_{q}(x);
(4.4.5)  r_{ψ}(U(y))φ_{q}(x) = ψ_{q}(y det(x))φ_{q}(x) = φ_{q}(x);
(4.4.6)  r_{ψ}(W)φ_{q}(x) = φ̂_{q}(x) = φ_{q}(x).

Next let us suppose that we are still in case (1) but q | N and q^{n} || N, so that

φ_{q} ( \begin{matrix}a & b \\ c & d \end{matrix} ) = \{ \begin{matrix}I_{Z_{q}}(a)I_{Z_{q}}(b)I_{q^{n}Z_{q}}(c)I_{Z_{q}}(d), if q ∉ Ξ; \\ \frac{1}{q}I_{Z_{q}}(a)I_{Z_{q}}(b)I_{q^{n−1}Z_{q}}(c)I_{Z_{q}}(d), if q ∈ Ξ. \end{matrix}

Note that

φ̂_{q} ( \begin{matrix}a & b \\ c & d \end{matrix} ) = \{ \begin{matrix}\frac{1}{q^{n}} I_{Z_{q}}(a)I_{q^{−n}Z_{q}}(b)I_{Z_{q}}(c)I_{Z_{q}}(d), if q ∉ Ξ; \\ \frac{1}{q^{n}} I_{Z_{q}}(a)I_{q^{−(n−1)}Z_{q}}(b)I_{Z_{q}}(c)I_{Z_{q}}(d), if q ∈ Ξ. \end{matrix}

Set

V (z) := ( \begin{matrix}1 & 0 \\ z & 1 \end{matrix} ) .

Then α is in the subgroup generated by matrices of the form D(a), U(y) and V (z) with a ∈ Z_{q}^{×}, y ∈ Z_{q} and z ∈ q^{n}Z_{q}. Now one checks immediately that the relations (4.4.4) and (4.4.5) continue to hold for such q. As for V (z), note that V (z) = D(−1)W U(z)W. Further, for z ∈ q^{n}Z_{q},

r_{ψ}(U(z))φ̂_{q} = φ_{q}.

Hence for such z,

r_{ψ}(V (z))φ_{q} = r_{ψ}(D(−1)W U(z)W)φ_{q} = r_{ψ}(D(−1)W U(z))φ̂_{q} = r_{ψ}(D(−1)W)φ̂_{q} = r_{ψ}(D(−1)) ˆφ̂_{q} = φ_{q}.

Thus case (1) is entirely verified. We now deal with case (2). In this case,

φ_{q} ( \begin{matrix}a & b \\ c & d \end{matrix} ) = \{ \begin{matrix}ε_{f,q}(d)I_{Z_{q}}(a)I_{Z_{q}}(b)I_{q^{n}Z_{q}}(c)I_{Z_{q}^{×}}(d), if q ∉ Ξ; \\ \frac{1}{q}ε_{f,q}(d)I_{Z_{q}}(a)I_{Z_{q}}(b)I_{q^{n−1}Z_{q}}(c)I_{Z_{q}^{×}}(d), if q ∈ Ξ. \end{matrix}

Thus

r_{ψ}(D(a))φ_{q}(x) = φ_{q}(ax) = ε_{f,q}(a)φ_{q}(x)

for a ∈ Z_{q}^{×} and r_{ψ}(U(y))φ_{q}(x) = ψ_{q}(y det(x))φ_{q}(x) = φ_{q}(x) for y ∈ Z_{q}.

It remains to consider the action of r_{ψ}(V (z)) on φ_{q} for z ∈ q^{n}Z_{q}. For this we need as before to compute the Fourier transform of φ_{q}. Suppose that cond(ε_{f,q}) = q^{m}Z_{q}, so that m ≤ n. Then

φ̂_{q} ( \begin{matrix}a & b \\ c & d \end{matrix} ) = \{ \begin{matrix}\frac{1}{q^{m+n}} ε_{f,q}^{−1}(q^{m}a)I_{q^{−m}Z_{q}^{×}}(a)I_{q^{−n}Z_{q}^{×}}(b)I_{Z_{q}}(c)I_{Z_{q}}(d), if q ∉ Ξ; \\ \frac{1}{q^{m+n}} ε_{f,q}^{−1}(q^{m}a)I_{q^{−m}Z_{q}^{×}}(a)I_{q^{−(n−1)}Z_{q}^{×}}(b)I_{Z_{q}}(c)I_{Z_{q}}(d), if q ∈ Ξ. \end{matrix}

Thus r_{ψ}(V (z))φ̂_{q} = φ̂_{q} in this case as well, and we see as above that r_{ψ}(V (z))φ_{q} = φ_{q}. □

We need the following lemma in order to study explicit theta lifts in both directions. For any q ∈ Σ, and for β ∈ GL_{2}(𝔸), we define

Φ_{q}(β) := ∫_{SL_{2}(Q_{q})} φ_{q}^{2}(α_{q}^{−1})F^{j}(βα_{q})d^{(1)}α_{q}.

Lemma 4.17. Let Σ′ denote the subset of Σ consisting of those primes q such that π_{f,q} ≃ π(μ_{1}, μ_{2}) is a ramified principal series representation with μ_{1} unramified and μ_{2} ramified of conductor exactly q^{n_{q}}, where q^{n_{q}}||N. Then for q ∈ Σ, the function Φ_{q}(β) is identically zero unless q ∈ Σ′. If q ∈ Σ′, then

Φ_{q}(β) = q^{−1/2}μ_{1}(q)^{−1}F^{j}(βγ_{q}),

where γ_{q} is the element of GL_{2}(𝔸) that is ( \begin{matrix}q & 0 \\ 0 & 1 \end{matrix} ) at q and 1 at all other places.

Proof. Let us write n instead of n_{q} for ease of notation. We suppose first that q ∈ Σ \ Σ′. In this case, π_{f,q} is either supercuspidal or a ramified special representation or a ramified principal series ≃ π(μ_{1}, μ_{2}) where μ_{1} and μ_{2} both have conductor dividing q^{n−1}. In any case, the central character ε_{f,q} has conductor dividing q^{n−1}. (See [Tu1] Prop. 3.4.) We claim then that

(4.4.7)  Φ_{q}(βu) = ε_{f,q}(d)Φ_{q}(β),

for u = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{q}(n − 1), where for any integer m ≥ 1, we define

Γ_{q}(m) := \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(Z_{q}) : c ≡ 0 mod q^{m}\} .

It suffices to verify (4.4.7) for γ a matrix in one of the three forms:

D(a, b) := ( \begin{matrix}a & 0 \\ 0 & b \end{matrix} ) , a, b, ∈ Z_{q}^{×}; U(y), y ∈ Z_{q}; and V (z), z ∈ q^{n−1}Z_{q}.

This follows from the following set of computations. First, let a, b ∈ Z_{q}^{×}. Then

Φ_{q}(β · D(a, b)) = ∫_{SL_{2}(Q_{q})} φ_{q}^{2}(α_{q}^{−1})F^{j}(β · D(a, b) · α_{q} · D(a, b)^{−1} · D(a, b))d^{(1)}α_{q}
= ε_{f,q}(b) ∫_{SL_{2}(Q_{q})} φ_{q}^{2}(D(a, b) · α_{q}^{−1} · D(a, b)^{−1})F^{j}(β · α_{q})d^{(1)}α_{q}
= ε_{f,q}(b) ∫_{SL_{2}(Q_{q})} φ_{q}^{2}(α_{q}^{−1})F^{j}(β · α_{q})d^{(1)}α_{q} = ε_{f,q}(b)Φ_{q}(β).

Next, let y ∈ Z_{q}. Then

Φ_{q}(β · U(y)) = ∫_{SL_{2}(Q_{q})} φ_{q}^{2}(α_{q}^{−1})F^{j}(β · U(y) · α_{q})d^{(1)}α_{q}
= ∫_{SL_{2}(Q_{q})} φ_{q}^{2}(α_{q}^{−1} · U(y))F^{j}(βα_{q})d^{(1)}α_{q}.

Suppose α_{q}^{−1} = ( \begin{matrix}a & b \\ c & d \end{matrix} ). Then

α_{q}^{−1} · U(y) = ( \begin{matrix}a & ay + b \\ c & cy + d \end{matrix} ) .

If φ_{q}^{2}(α_{q}^{−1}) ≠ 0, then a, b, d ∈ Z_{q} and c ∈ q^{n−1}Z_{q}. Hence cy +d ≡ d mod q^{n−1}. Since the conductor of ε_{f,q} divides q^{n−1}, it follows that φ_{q}^{2}(α_{q}^{−1}U(y)) = φ_{q}^{2}(α_{q}^{−1}) for all α_{q}, and consequently Φ_{q}(β · U(y)) = Φ_{q}(β). Finally, let z ∈ q^{n−1}Z_{q}. Then

Φ_{q}(β · V (z)) = ∫_{SL_{2}(Q_{q})} φ_{q}^{2}(α_{q}^{−1})F^{j}(β · V (z) · α_{q})d^{(1)}α_{q}
= ∫_{SL_{2}(Q_{q})} φ_{q}^{2}(α_{q}^{−1} · V (z))F^{j}(βα_{q})d^{(1)}α_{q}.

But

α_{q}^{−1}V (z) = ( \begin{matrix}a + bz & b \\ c + dz & d \end{matrix} ) .

Since z ∈ q^{n−1}Z_{q}, one finds that φ_{q}^{2}(α_{q}^{−1}V (z)) = φ_{q}^{2}(α_{q}^{−1}) for all α_{q}. This proves (4.4.7). But now by Casselman’s theorem, we see that Φ_{q}(β) must be identically zero for such q.

We now turn to q ∈ Σ′. In this case, one cannot argue as above since ε_{f,q} has conductor q^{n}. However the argument above shows that Φ_{q} is right invariant by V (z) for z ∈ q^{n−1}Z_{q}, and by U(y) for y ∈ qZ_{q}, and transforms by ε_{f,q}(b) under the right action of D(a, b). We conclude that if u lies in the subgroup

\{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(Z_{q}) : a, d ∈ Z_{q}^{×}, b ∈ qZ_{q}, c ∈ q^{n−1}Z_{q}\} ,

then Φ_{q}(β · u) = ε_{f,q}(d(u))Φ_{q}(β). By Casselman’s theorem, we see that

Φ_{q}(βγ_{q}^{−1}) = c̃ · F^{j}(β)

for some scalar c̃. We now compute the value of c̃. Letting Γ_{q}^{(1)}(m) := Γ_{q}(m) ∩ SL_{2}(Q_{q}), note that

Φ_{q}(β) = \frac{1}{q} ∫_{Γ_{q}^{(1)}(n−1)} ε_{f,q}(d(α_{q}^{−1}))F^{j}(βα_{q})d^{(1)}α_{q}.

Let us first suppose that n ≥ 2. Then the collection

V (x) = ( \begin{matrix}1 & 0 \\ x & 1 \end{matrix} ) , x ∈ q^{n−1}Z_{q}/q^{n}Z_{q}.

is a set of coset representatives for Γ_{q}^{(1)}(n − 1)/Γ_{q}^{(1)}(n). Hence

c̃ · F^{j}(βγ_{q}) = Φ_{q}(β) = \frac{1}{q} ∑_{x∈q^{n−1}Z_{q}/q^{n}Z_{q}} ∫_{Γ_{q}^{(1)}(n)} ε_{f,q}(d(α_{q}^{−1}V (x)))F^{j}(βV (x)α_{q})d^{(1)}α_{q}

= \frac{1}{q} ∑_{x∈q^{n−1}Z_{q}/q^{n}Z_{q}} ∫_{Γ_{q}^{(1)}(n)} ε_{f,q}(d(α_{q}^{−1}))F^{j}(βV (x))ε_{f,q}(d(α_{q}))d^{(1)}α_{q}

(4.4.8)  = \frac{1}{q} vol(U′_{q}^{(1)}) ∑_{x∈q^{n−1}Z_{q}/q^{n}Z_{q}} F^{j}(βV (x)).

To find the value of c̃ we may substitute β = 1 and compute in a convenient model for the local representation π_{f,q} ≃ π(μ_{1}, μ_{2}). We use the standard model of the induced representation V (μ_{1}, μ_{2}), and denote by f_{q} a new vector in this representation, normalized so that f_{q}(1) = 1. Then (see [SR], Prop. 2.1.2)

f_{q}(γ_{q}) = μ_{1}(q)^{1−n}|q|_{q}^{1/2}

while

f_{q} ( \begin{matrix}1 & 0 \\ x & 1 \end{matrix} ) = \{ \begin{matrix}μ_{1}(q)^{−n}, if v_{q}(x) ≥ n; \\ 0, if v_{q}(x) < n. \end{matrix}

It follows that

c̃ = \frac{1}{q}μ_{1}(q)^{−1}|q|_{q}^{−1/2} vol(U′_{q}^{(1)}) = q^{−1/2}μ_{1}(q)^{−1} vol(U′_{q}^{(1)}).

If on the other hand n = 1, then the matrices V (x) with x ∈ Z_{q}/qZ_{q} along with W form a set of coset representatives for Γ_{q}^{(1)}/Γ_{q}^{(1)}(1). Again we can use the standard model of the induced representation to compute the value of c̃. However, since f_{q}(W) = 0 (see [SR], Prop. 2.1.2), the expression for c̃ remains the same in this case too.

□

Definition 4.18. For Ξ ⊂ Σ, we set F_{Ξ}^{j}(g) = F^{j}(g · ∏_{q∈Ξ} γ_{q}), where γ_{q} is as in Lemma 4.17 above.

Proposition 4.19.

θ_{φ^{Ξ}}^{t}(F^{j} × \overline{F_{Ξ}^{j}}) = C_{1}^{Ξ} · F^{0,♯},

where

(4.4.9)  C_{1}^{Ξ} := \{ \begin{matrix}0, if Ξ ⊄ Σ′; \\ (4π)^{−(j−1)} \frac{Γ(k+j)}{Γ(k)} vol(U′^{(1)}) · ⟨F^{j}, F^{j}⟩ · ∏_{q∈Ξ}(q^{−1/2}μ_{1}(q)) if Ξ ⊆ Σ′, \end{matrix}

and F^{0,♯} is the unique form in π characterized by (i) If q ∤ N, then F^{0,♯}(gu) = F^{0,♯}(g) for u ∈ GL_{2}(Z_{q}).

(ii) If q | N, then F^{0,♯}(gu) = ε_{f,q}(a)F^{0,♯}(g) for u = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{q}(n_{q}).

(iii) Let a ∈ R^{×}, a_{∞} := d(a) ∈ GL_{2}(R), κ_{θ} = ( \begin{matrix}cos θ & − sin θ \\ sin θ & cos θ \end{matrix} ) ∈ SO_{2}(R). Let (1, a_{∞}κ_{θ}) denote

the element of GL_{2}(𝔸) which is 1 at all finite places and a_{∞}κ_{θ} at the infinite place. Then

W_{F^{0,♯},ψ}(1, a_{∞}κ_{θ}) = a^{k/2}e^{−2πa}e^{ikθ}I_{R^{+}}(a).

Here W_{·,ψ} denotes as usual the ψ-Whittaker coefficient and ⟨F^{j}, F^{j}⟩ denotes the Petersson inner product:

⟨F^{j}, F^{j}⟩ = \frac{1}{2} ∫_{PGL_{2}(Q)\PGL_{2}(𝔸)} F^{j}(β)\overline{F^{j}(β)}d^{×}β.

Proof. Let F′ := θ_{φ^{Ξ}}^{t}(F^{j} × \overline{F_{Ξ}^{j}}). We first show that F′ = C_{1}^{Ξ} · F^{0,♯} for some constant C_{1}^{Ξ}. Note that for u ∈ U′ and κ_{θ} ∈ SO_{2}(R), by Lemmas 4.15 and 4.16,

(4.4.10) F′(guκ_{θ}) = ∫_{O(V)(Q)\O(V)(𝔸)} ∑_{x∈V (Q)} r_{ψ}(guκ_{θ}, h · (u, 1))φ^{Ξ}(x)(F^{j} × \overline{F_{Ξ}^{j}})(h · (u, 1))dh
= e^{ikθ} ∏_{q|N_{ε_{f}}} ε_{f,q}(a(u_{q}))ε_{f,q}(d(u_{q})^{−1}) · ε_{f,q}(d(u_{q}))F′(g)

= e^{ikθ} ∏_{q|N_{ε_{f}}} ε_{f,q}(a(u_{q}))F′(g).

Since θ_{ψ}^{t}(π ⊗ π̄) = π, it follows by Casselman’s theorem that F′ = C_{1}^{Ξ} · F^{0,♯} for some scalar C_{1}^{Ξ}. Clearly, C_{1}^{Ξ} is just the first Fourier coefficient of F′. To evaluate C_{1}^{Ξ}, we compute the Whittaker coefficients of F′. As in [Wa] Sec. 3.2.1,

W_{F′,ψ}(g) = \frac{1}{2} ∫_{PGL_{2}(Q)\PGL_{2}(𝔸)} Ψ(g, β)\overline{F_{Ξ}^{j}}(β)d^{×}β,

where

Ψ(g, β) = ∫_{GL_{2}(𝔸)^{det(g)}} r_{ψ}(g, (α, 1))φ^{Ξ}(1)F^{j}(βα)d^{(1)}α.

Note that

Ψ(1, β) = ∫_{SL_{2}(𝔸)} r_{ψ}(1, (α, 1))φ^{Ξ}(1)F^{j}(βα)d^{(1)}α
= ∫_{SL_{2}(𝔸)} φ^{Ξ}(α^{−1})F^{j}(βα)d^{(1)}α.

This integral can be computed one place at a time since both F^{j} and φ^{Ξ} are pure tensors. We first consider finite primes q such that q ∉ Ξ. In this case, if φ_{q}(α_{q}^{−1}) ≠ 0, then α_{q}^{−1} ∈ U′_{q}. Hence α_{q} ∈ U′_{q} as well. If further q ∤ N_{ε_{f}}, then φ_{q}(α_{q}^{−1}) = 1 and F^{j}(βα_{q}) = F^{j}(β). On the other hand, if q | N_{ε_{f}}, then φ_{q}(α_{q}^{−1}) = ε_{f,q}(d(α_{q})^{−1}) and F^{j}(βα_{q}) = ε_{f,q}(d(α_{q}))F^{j}(β), so that in any case, for q ∉ Ξ, we have

∫_{SL_{2}(Q_{q})} φ_{q}^{Ξ}(α_{q}^{−1})F^{j}(βα_{q})d^{(1)}α_{q} = vol(U′_{q}^{(1)}) · F^{j}(β).

For q ∈ Ξ, it follows from Lemma 4.17 that

∫_{SL_{2}(Q_{q})} φ_{q}^{Ξ}(α_{q}^{−1})F^{j}(βα_{q})d^{(1)}α_{q} = \{ \begin{matrix}0, if q ∉ Σ′; \\ vol(U′_{q}^{(1)}) · q^{−1/2}μ_{1}^{−1}(q)F^{j}(βγ_{q}), if q ∈ Σ′. \end{matrix}

Finally, the computation of the local integral at the infinite place can be found in [Xue], Prop. 4.3.4. Accounting for our different choice of measures, this contribution equals e^{−2π}(4π)^{−(j−1)}Γ(k + j)/Γ(k). Puting together the local computations, we find

Ψ(1, β) = \{ \begin{matrix}0, if Ξ ⊄ Σ′; \\ e^{−2π} · (4π)^{−(j−1)} \frac{Γ(k+j)}{Γ(k)} · vol(U′^{(1)}) · ∏_{q∈Ξ}(q^{−1/2}μ_{1}^{−1}(q)) · F_{Ξ}^{j}(β), if Ξ ⊂ Σ′. \end{matrix} .

Thus C_{1}^{Ξ} = 0 unless Ξ ⊆ Σ′ and in that case,

C_{1}^{Ξ} = e^{2π}W_{F′,ψ}(1) = (4π)^{−(j−1)} vol(U′^{(1)})\frac{Γ(k + j)}{Γ(k)} · ∏_{q∈Ξ}(q^{−1/2}μ_{1}^{−1}(q))⟨F_{Ξ}^{j}, F_{Ξ}^{j}⟩

= (4π)^{−(j−1)} \frac{Γ(k + j)}{Γ(k)} · vol(U′^{(1)})⟨F^{j}, F^{j}⟩ · ∏_{q∈Ξ}(q^{−1/2}μ_{1}^{−1}(q)). □

Proposition 4.20.

θ_{φ}(\overline{F^{0,♯}}) = C_{2}^{Ξ} · (\overline{F^{j}} × F_{Ξ}^{j}),

where

(4.4.11)  C_{2}^{Ξ} = \{ \begin{matrix}0, if Ξ ⊄ Σ′; \\ \frac{(4π)^{j+1}}{Γ(j+1)} ℑ(τ)^{ℓ} vol(U′^{(1)}) ∏_{q∈Σ′}(q^{−1/2}μ_{1}^{−1}(q)) if Ξ ⊆ Σ′. \end{matrix}

(Recall that Σ′ was defined in Lemma 4.17.)

Proof. By a calculation as in (4.4.10) above and another application of Casselman’s theorem, we have θ_{φ}(\overline{F^{0,♯}}) = C_{2}^{Ξ} ·(\overline{F^{j}} ×F_{Ξ}^{j}) for some constant C_{2}^{Ξ}. To compute C_{2}^{Ξ}, one studies the theta lift in the opposite direction and uses the seesaw principle. Indeed, the seesaw principle and Proposition 4.19 imply that

C_{2}^{Ξ}⟨F^{j}, F^{j}⟩^{2} = ⟨θ_{φ}(\overline{F^{0,♯}}), \overline{F^{j}} × F_{Ξ}^{j}⟩ = ⟨\overline{F^{0,♯}}, \overline{θ_{φ}^{t}(F^{j} × \overline{F_{Ξ}^{j}})}⟩ = C_{1}^{Ξ}⟨F^{0,♯}, F^{0,♯}⟩.

i.e., C_{2}^{Ξ} = C_{1}^{Ξ}⟨F^{0,♯}, F^{0,♯}⟩/⟨F^{j}, F^{j}⟩^{2}. But (see Lemma 4.11),

⟨F^{j}, F^{j}⟩/⟨F^{0,♯}, F^{0,♯}⟩ = ℑ(τ)^{−ℓ}(4π)^{−2j}Γ(j + 1)Γ(k + j)/Γ(k).

(The term ℑ(τ)^{−ℓ} appears since F^{0} and F^{0,♯} are normalized differently: the former is the adelic form associated to f and the base point τ, while the latter uses the base point i. To translate from one to other involves picking an element γ ∈ SL_{2}(R) such that γi = τ and one checks that ⟨F^{0}, F^{0}⟩/⟨F^{0,♯}, F^{0,♯}⟩ = j(γ, i)^{2ℓ} = ℑ(τ)^{−ℓ}.) The proposition now follows by using the value of C_{1}^{Ξ} from Prop. 4.19.

□

We now make the following key definition, namely that of the Schwartz function in the explicit theta correspondence.

Definition 4.21. The explicit Schwartz function φ is defined by φ := ⊗_{q}φ_{q}, where φ_{∞} is as in (4.4.3) and for finite primes q, the φ_{q} are as below:

(i) If q ∤ N, then φ_{q} = I_{M_{0}(N)⊗Z_{q}} = I_{M_{2}(Z_{q})}.

(ii) If q | N, then φ_{q} = φ_{q}^{1} for q ∉ Σ′ and φ_{q} := φ_{q}^{1} − φ_{q}^{2} for q ∈ Σ′. Recall that φ_{q}^{1} and φ_{q}^{2} were defined previously in (4.4.1) and (4.4.2) respectively and Σ′ was defined in Lemma 4.17.

The following lemma which will be used in the next section is an easy consequence of the fact that η is of type (c, 𝔑, ε_{f}^{−1}).

Lemma 4.22. For q ∈ Σ′, fix an isomorphism K_{q} ≃ Q_{q} × Q_{q} such that via this identification the

embedding ξ_{q} : K_{q} ↪ M_{2}(Q_{q}) is conjugate by an element of U′_{q} to the embedding (a, b) ↦ ( \begin{matrix}a & 0 \\ 0 & b \end{matrix} ). Let

\overline{η′_{q}} = (η_{1}, η_{2}) via this identification. Then (1) η_{1} is unramified and η_{2} is ramified. (ii) η_{2}μ_{2}^{−1} is unramified.

4.5. Seesaw duality and the Siegel-Weil formula. Let V_{1} = K (viewed as a subspace of V via ξ) and let V_{2} = V_{1}^{⊥}. Then

GO(V_{1})^{0} ≃ GO(V_{2})^{0} ≃ K^{×},
H := G(O(V_{1}) × O(V_{2}))^{0} = G(K^{×} × K^{×}),

and via this identification the map δ : K^{×} × K^{×} → H is

δ(α, β) = (αβ^{−1}, α(β^{ρ})^{−1}).

Since \overline{η′}(α)η′(β) = \overline{η′}(αβ^{−1}), the character (\overline{η′}, η′) of K^{×} × K^{×} is the pullback via δ of the character η := (\overline{η′}, 1) on H. Suppose that

φ_{q} = ∑_{i_{q}∈I_{q}} φ_{1}^{i_{q}} ⊗ φ_{2}^{i_{q}} ∈ S(V_{1} ⊗ Q_{q}) ⊗ S(V_{2} ⊗ Q_{q}).

Then by an application of seesaw duality for the seesaw pair

[figure: GL_{2} × GL_{2}; GO(V); _{GL_{2}}; G(O(V_{1}) × O(V_{2}))]

we have (as in [HK2] (14.5)),

∫_{H(Q)\H(𝔸)} θ_{ψ,φ}(\overline{F^{0,♯}})|_{H(𝔸)}(h)η(h)d^{×}h
= ∫_{GL_{2}(Q)𝔸^{×}\GL_{2}(𝔸)} \overline{F^{0,♯}}(g) · θ_{φ}^{t}(η)|_{GL_{2}(𝔸)}(g)dg
(4.5.1)  = ∫_{GL_{2}(Q)𝔸^{×}\GL_{2}(𝔸)} \overline{F^{0,♯}}(g) · ∑ ; i=(i_{q})∈I=∏_{q} I_{q}θ_{⊗_{q}φ_{1}^{iq}}^{t} (\overline{η′})(g)θ_{⊗_{q}φ_{2}^{iq}}^{t} (1)(g)dg.

Here θ^{t}(\overline{η′}) and θ^{t}(1) are defined as follows. Set

GL_{2}(𝔸)^{K} := {g ∈ GL_{2}(𝔸) : det(g) ∈ N_{K}(𝔸_{K}^{×})} .

For g ∈ GL_{2}(𝔸)^{K}, ς ∈ S(V_{1}(𝔸)) and h ∈ 𝔸_{K}^{×} such that det(g) = N_{K}(h),

θ_{ς}^{t}(\overline{η′})(g) := ∫_{K^{(1)}\K_{𝔸}^{(1)}} ∑_{x∈V_{1}} r_{ψ}(g, hh_{1})ς(x)\overline{η′}(hh_{1})d^{(1)}h_{1}.

One then extends the definition to the index 2 subgroup GL_{2}(Q) · GL_{2}(𝔸)^{K} of GL_{2}(𝔸) by requiring it to be left invariant by GL_{2}(Q). Finally, one extends it by zero outside this index two subgroup. The theta lift θ^{t}(1) is defined similarly with \overline{η′} replaced by the trivial character and V_{1} replaced by V_{2}. Here the measure d^{(1)}h_{1} is chosen such that it lifts to a Haar measure on K_{𝔸}^{(1)} and vol(K^{(1)} \ K_{𝔸}^{(1)}) = 1.

Now, by the Siegel-Weil formula, the theta lift θ^{t}(1) is an Eisenstein series. Unfolding this Eisenstein series by the standard Rankin-Selberg method, one finds that the integral in (4.5.1) above is equal to the expression I(φ, ξ), where (defining Φ^{s} as in [P] Prop. 3.1.),

I(φ, ξ) := ζ(2)^{−1} ∫_{𝔸_{Q}^{×}} ∫_{K_{0}} Wψ̄(\overline{F^{0,♯}})(d(a)k) ∑ ; i=(i_{q})∈I=∏_{q} I_{q} W_{ψ}(θ_{⊗_{q}φ_{1}^{iq}}^{t} (\overline{η′}))(d(a)k)Φ_{⊗_{q}φ_{2}^{iq}}^{s} (d(a)k)(1)|a|^{−1}d^{×}adk,

Here K_{0} = ∏_{q} GL_{2}(Z_{q}) × SO_{2}(R)), the measure dk is a product of local Haar measures such that vol(GL_{2}(Z_{q})) = 1 and vol(SO_{2}(R)) = 2π, and the factor ζ(2)^{−1} accounts for the change in measure normalization. We now state two propositions that will be useful in computing the integral above.

We note first that Wψ̄(\overline{F^{0,♯}}) = \overline{W_{ψ}(F^{0,♯})} and W_{ψ}(F^{0,♯}) = ∏_{v} W_{ψ,v}(F^{0,♯}) where W_{ψ,v}(F^{0,♯}) is normalized to take value 1 on the identity matrix in GL_{2}(Z_{q}) for finite q and W_{ψ,∞}(F^{0,♯})(d(a)) = e^{−2πa}a^{\frac{k}{2}} I_{R^{+}}(a). The proposition below (which is simply copied from [SR] Sec. 2.4 taking into account that F^{0,♯} transforms by the central character of the upper left entry at ramified places as opposed to the lower right entry as

in loc. cit.) lists the values of W_{ψ,q}(F^{0,♯}) on matrices of the form d(a) := ( \begin{matrix}a & 0 \\ 0 & 1 \end{matrix} ).

Proposition 4.23. Let a ∈ Q_{q}^{×}. Then W_{ψ,q}(F^{0,♯})(d(a)) is equal to

(i) |a|^{1/2} (∑_{r+s=v_{q}(a)} μ_{1}(q)^{r}μ_{2}(q)^{s}) I_{Z_{q}}(a) if π_{f,q} ≃ π(μ_{1}, μ_{2}) is an unramified principal series representation.

(ii) |a|μ(a)I_{Z_{q}}(a), if π_{f,q} ≃ St(μ) is a special representation with μ unramified.

(iii) I_{Z_{q}^{×}}(a), if π_{f,q} ≃ St(μ) is a special representation with μ ramified.

(iv) |a|^{1/2}μ_{2}(a)I_{Z_{q}}(a), if π_{f,q} ≃ π(μ_{1}, μ_{2}) is a ramified principal series representation with μ_{1} unrami-

fied and μ_{2} ramified.

(v) ε_{f,q}(a)I_{Z_{q}^{×}}(a), if π_{f,q} ≃ π(μ_{1}, μ_{2}) is a ramified principal series representation with both μ_{1} and μ_{2} ramified, or if π_{f,q} is supercuspidal.

For simplicity, in our local calculations below, we will simply write W_{F} for W_{ψ̄,q}(\overline{F^{0,♯}}). The following proposition follows from the discussion in [P] §3.3.

Proposition 4.24. The Whittaker function W_{ψ}(θ_{⊗_{q}ϑ_{q}}^{t}(\overline{η′})) factors as

W_{ψ}(θ_{⊗_{q}ϑ_{q}}^{t}(\overline{η′})) = \frac{1}{h_{K}} ∏_{q} W_{Θ,ϑ_{q}},

where for any prime q, either finite or infinite,

W_{Θ,ϑ_{q}}(d(a)) = ∫_{K_{q}^{(1)}} ϑ_{q}(a(hh′)^{−1})\overline{η′_{q}}(hh′)dh
(4.5.2)  = |a|_{q}^{1/2} ∫_{K_{q}^{1}} ϑ_{q}((hh′)^{ρ})\overline{η′_{q}}(hh′)dh,

for any h′ such that N(h′) = a. (Here the Haar measure dh on K_{v}^{(1)} is chosen such that vol(K_{∞}^{(1)}) = 1 and for finite primes q, vol(O_{K} ⊗ Z_{q})^{(1)} = 1.) Also,

Φ_{⊗_{ςq}}^{s} (d(a)) = |a|^{s} ∏_{q} ς_{q}(0).

More generally, suppose j_{q} : K_{q} → V_{q} is an embedding of quadratic spaces, where K_{q} = K ⊗ Q_{q} and V_{q} = V (Q_{q}). For ς ∈ S(V_{q}) = S(K_{q})) ⊗ S(K_{q}^{⊥}), write ς = ∑_{i} ς_{1,i} ⊗ ς_{2,i} and define

(4.5.3)  I(ς, j_{q}) = ∑_{i} ∫_{Q_{q}^{×}} ∫_{K_{0,q}} W_{F} (d(a)k)W_{Θ,ς_{1,i}}(d(a)k)Φ_{ς_{2,i}}^{s}(d(a)k)|a|^{−1}d^{×}adk

Since W_{Θ,ς_{1,i}} · Φ_{2,i}^{s}(·) is bilinear in (ς_{1,i}, ς_{2,i}), the expression on the right in (4.5.3) is independent of the decomposition ς = ∑_{i} ς_{1,i} ⊗ ς_{2,i}. In this notation, we have

(4.5.4)  I(φ, ξ) = \frac{ζ(2)^{−1}}{h_{K}} ∏_{q<∞} I(φ_{q}, ξ_{q}) · I(φ_{∞}, ξ_{∞}).

Thus to compute I(φ, ξ) it suffices to compute I(φ_{q}, ξ_{q}) for all q. However, for finite primes q, it is easier to compute I(φ_{q}, ξ′_{q}) for a modified embedding ξ′_{q} which is defined by

ξ′_{q}(x) = u_{q}^{−1}ξ_{q}(x)u_{q}

for some suitable choice of u_{q} ∈ U′_{q}. If φ′_{q} is the Schwartz function defined by

φ′_{q}(x) = φ_{q}(u_{q}^{−1}xu_{q}),

then it is immediate that

I(φ′_{q}, ξ_{q}) = I(φ_{q}, ξ′_{q}).

Define φ′ by φ′ = (⊗_{q}φ′_{q}) ⊗ φ_{∞}.

Proposition 4.25. Suppose that the u_{q} ∈ U′_{q} have been chosen such that for all q ∈ Σ′, ξ′_{q} is given on K_{q} = Q_{q} × Q_{q} by

ξ′_{q}(a, b) = ( \begin{matrix}a & 0 \\ 0 & b \end{matrix} ) .

Then

∫_{H(Q)\H(𝔸)} θ_{φ′}(\overline{F^{0,♯}})|_{H(𝔸)}(h)η(h)d^{×}h = \frac{(4π)^{j+1}ℑ(τ)^{ℓ}}{Γ(j + 1)} vol(U′^{(1)}) · ∏_{q∈Σ′}(1 − μ_{1}^{−1}(q)η_{1}(q)q^{−1/2}) · |L_{η′,ξ}(F^{j})^{2}|.

Proof. Let u ∈ GL_{2}(𝔸_{f}) be the element whose coordinate at q is u_{q}. Observe that φ′ = r_{ψ}(1, (u, u))φ. Hence θ_{φ′}(\overline{F^{0,♯}})(h) = θ_{φ}(\overline{F^{0,♯}})(h · (u, u)), and

∫_{H(Q)\H(𝔸)} θ_{φ′}(\overline{F^{0,♯}})|_{H(𝔸)}(h)η(h)d^{×}h
= θ_{φ}(\overline{F^{0,♯}})|_{H(𝔸)}(h · (u, u))η(h)d^{×}h
= ∑_{Ξ⊂Σ′}(−1)^{|Ξ|}θ_{φ^{Ξ}}(\overline{F^{0,♯}})|_{H(𝔸)}(h · (u, u))η(h)d^{×}h

= ∑_{Ξ⊂Σ′}(−1)^{|Ξ|}C_{2}^{Ξ} ∫_{K^{×}×K^{×}\𝔸_{K}^{×}×𝔸_{K}^{×}} ((\overline{F^{j}} × F_{Ξ}^{j})(αu, βu) · (\overline{η′} × η′)(α, β)d^{×}αd^{×}β
= ∑_{Ξ⊂Σ′}(−1)^{|Ξ|}C_{2}^{Ξ} · \overline{L_{η′,ξ}(F^{j}(·u))} · L_{η′,ξ}(F_{Ξ}^{j}(·u)).

But setting α_{q} := (q^{−1}, 1) ∈ K_{q}^{×}, α_{Ξ} := ∏_{q∈Ξ} α_{q} and γ_{Ξ} := ∏_{q∈Ξ} γ_{q}, we have ξ_{𝔸}(α_{Ξ}) · uγ_{Ξ}u^{−1} = 1 and

L_{η′,ξ}(F_{Ξ}^{j}(·u)) = ∫_{K^{×}\𝔸_{K}^{×}} F^{j}(ξ_{𝔸}(x)uγ_{Ξ})η′(x)d^{×}x = ∫_{K^{×}\𝔸_{K}^{×}} F^{j}(ξ_{𝔸}(xα_{Ξ})uγ_{Ξ})η′(xα_{Ξ})d^{×}x

= η′(α_{Ξ})L_{η′,ξ}(F^{j}(·u)) = (∏_{q∈Ξ} η_{1}(q)) · L_{η′,ξ}(F^{j}(·u)).

Since F^{j}(·u) = F^{j}(·)ω_{f}(u), the proposition follows by using the value of C_{2}^{Ξ} from (4.4.11). □

We record the following corollary, which follows from the above proposition and the preceding discussion.

Corollary 4.26.

I(φ′, ξ) = \frac{(4π)^{j+1}ℑ(τ)^{ℓ}}{Γ(j + 1)} vol(U′^{(1)}) · ∏_{q∈Σ′}(1 − μ_{1}^{−1}(q)η_{1}(q)q^{−1/2}) · |L_{η′,ξ}(F^{j})^{2}|.

Applying (4.5.4) (with φ replaced by φ′), we see that to compute |L_{η′,ξ}(F^{j})|^{2}, it suffices to compute I(φ′_{q}, ξ_{q}) = I(φ_{q}, ξ′_{q}) for convenient choices of ξ′_{q} satisfying the hypotheses of the lemma above. This is the content of the next section.

4.6. Local zeta integrals. To handle the local computations, it will be useful to set up the following notation. Define

J(ς, ϑ) := ∫_{Q_{q}^{×}} W_{F} (d(a))W_{Θ,ς}(d(a))Φ_{ϑ}^{s}(d(a))|a|^{−1}d^{×}a,

and for α ∈ GL_{2}(Q_{q}),

J(ς, ϑ, α) := ∫_{Q_{q}^{×}} W_{F} (d(a)α)W_{Θ,ς}(d(a))Φ_{ϑ}^{s}(d(a))|a|^{−1}d^{×}a.

We first dispose the simple case q = ∞.

Proposition 4.27. For q = ∞, we have

I(φ_{∞}, ξ_{∞}) = (2π) · (4π)^{−(k+j)}Γ(k + j).

Proof. One sees easily that I(φ_{∞}, j_{∞}) = J(ς, ϑ), where

ς(u) = \overline{u}^{l}e^{−2π⟨u,u⟩}

and

ϑ(v) = p_{j}(4π⟨v, v⟩)e^{−2π⟨v,v⟩}.

Thus Φ_{ς_{1}}^{s}(d(a)) = |a|^{s}ϑ(0). Taking h′ = a^{1/2} in (4.5.2), we find

W_{Θ,ς}(d(a)) = I_{R^{+}}(a)|a|^{1/2} ∫_{K_{∞}^{(1)}} ς(a^{1/2}h^{−1})h^{−ℓ}dh
= a^{\frac{ℓ+1}{2}} e^{−2πa}I_{R^{+}}(a) = a^{\frac{ℓ+1}{2}} e^{−2πa}I_{R^{+}}(a).

Thus I(φ_{∞}, ξ_{∞}) = 2π · ∫_{R^{×}} a^{\frac{k}{2}} e^{−2πa} · a^{\frac{ℓ+1}{2}} e^{−2πa} · |a|^{s−1}I_{R^{+}}(a)d^{×}a and

I(φ_{∞}, ξ_{∞})|_{s=1/2} = 2π · ∫_{0}^{∞} a^{\frac{k+ℓ}{2}} e^{−4πa}d^{×}a = (2π) · (4π)^{−(k+j)}Γ(k + j).
□

Next let q be a finite prime, and denote by 𝔬_{q} and 𝔯_{q} the maximal orders in K_{q} and Q_{q} respectively. We split the calculations into several cases:

I : q ∤ cNd_{K}.

II : q | c.

III : q^{n_{q}}||N, with n_{q} ≥ 2.

IV : q||N, q ∤ d_{K}. V : q||N, q | d_{K}. VI : q | d_{K}, q ∤ N.

For the rest of this section, we simply write I for I(φ′, ξ_{q}) = I(φ, ξ′_{q}).

4.6.1. Case I: q ∤ cNd_{K}. In this case all the data is unramified and we have by a standard computation:

I = L_{q}(π̄_{f}, πη̄, s)L_{q}(2s, ε_{K})^{−1}.

4.6.2. Case II: q | c. Write 𝔬_{q} = Z_{q} + Z_{q}ϖ, where tr(ϖ) = 0. Let ϖ^{2} = u. We may assume that

ξ′_{q}(ϖ) = ( \begin{matrix}0 & 1/q^{r} \\ uq^{r} & 0 \end{matrix} ), where q^{r}||c. Set 𝔧_{q} := ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ). For 0 ≤ i, j ≤ q^{r} − 1, set

ς_{i,j} = I_{Z_{q}+(q^{r}Z_{q}+i+ \frac{j}{q^{r}} )ϖ} ϑ_{i,j} = I_{(Z_{q}+(q^{r}Z_{q}+i+ \frac{j}{q^{r}} )ϖ)𝔧_{q}}.

Then

φ_{q} = ∑_{i,j} ς_{i,j} ⊗ ϑ_{i,j}.

Since W_{F} and φ_{q} are invariant under GL_{2}(Z_{q}), it follows that

I = ∑_{i,j} J(ς_{i,j}, ϑ_{i,j}) = J(ς_{0,0}, ϑ_{0,0}).

Now,

W_{Θ,ς_{0,0}}(d(a)) = ∫_{Q_{q}^{×}} ς_{0,0}(t, at^{−1})η_{1}(at^{−1})η_{2}(t)d^{×}t = ∫_{v_{q}(t−at^{−1})≥r^{0≤vq (t)≤vq (a)}} η_{1}(at^{−1})η_{2}(t)d^{×}t.

Suppose v_{q}(a) ≥ 2r − 1. Then either v_{q}(t) ≥ r or v_{q}(at^{−1}) ≥ r. In this case, v_{q}(t − at^{−1}) ≥ r ⟺ both v_{q}(t) ≥ r and v_{q}(at^{−1}) ≥ r. For such a then, the region of integration in the last integral above is unchanged if a is replaced by ua for any u ∈ Z_{q}^{×}. Thus W_{Θ,ς_{0,0}}(d(au)) = η_{1}(u)W_{Θ,ς_{0,0}}(a). Since W_{F} (d(au)) = W_{F} (d(a)), by picking u such that η_{1}(u) ≠ 1, we see that

∫_{v_{q}(a)≥2r−1} W_{F} (d(a))W_{Θ,ς_{0,0}}(d(a))|a|^{s−1}d^{×}a = 0.

So we may restrict attention to a such that 0 ≤ v_{q}(a) ≤ 2r − 2, and let t be in the region of integration above. Since either v_{q}(t) ≤ r − 1 or v_{q}(at^{−1}) ≤ r − 1, we see that v_{q}(t − at^{−1}) ≥ r is only possible if v_{q}(t) = v_{q}(at^{−1}). This implies that v_{q}(a) must be even. Suppose that v_{q}(a) = 2m ≤ 2r − 2 so that m ≤ r − 1, and v_{q}(t) = m. Write a = q^{2m}u, t = q^{m}v with u, v ∈ Z_{q}^{×}. The condition v_{q}(t − at^{−1}) ≥ r then translates to v_{q}(v^{2} − u) ≥ r − m, and η_{1}(at^{−1})η_{2}(t) = η_{1}(q^{m}uv^{−1})η_{2}(q^{m}v) = ε_{f,q}(q)^{m}η_{1}(uv^{−2}) since η_{1}η_{2} = ε_{f,q} is unramified. Then for m fixed,

∫_{v_{q}(a)=m} W_{F} (d(a))W_{Θ,ς_{0,0}}(d(a)|a|^{s−1}d^{×}a = constant · ∫ ∫_{u≡v^{2} mod q^{r−m}^{u,v∈Z^{×}_{q}}} η_{1}(uv^{−2})d^{×}vd^{×}u.

Suppose m > 0. Since the conductor of η_{1} is q^{r}, there exists α ∈ Z_{q}^{×}, α ≡ 1 mod q^{r−m} such that η_{1}(α) ≠ 1. Then for v fixed the integral over u is seen to be zero by making a change of variables u ↦ αu. Thus we are reduced to considering only the case m = 0, and

I = vol((u, v) ∈ Z_{q}^{×} × Z_{q}^{×}, u ≡ v^{2} mod q^{r}) = \frac{1}{q^{r−1}(q − 1)} = \frac{1}{q^{r}} ζ_{K,q}(1) · L_{q}(π̄_{f}, πη̄′, s)L_{q}(2s, ε_{K})^{−1}|_{s=1/2}.

Here ζ_{K,q}(1) = (1 − \frac{1}{q})^{−2} if q is split in K and equal to (1 − \frac{1}{q^{2}} ) if q is inert in K.

4.6.3. Case III: q^{n}||N with n ≥ 2. In this case, q is split in K i.e. q = 𝔮𝔮̄ and K ⊗ Q_{q} ≃ Q_{q} × Q_{q} corresponding to the completions at 𝔮 and 𝔮̄ respectively. We suppose that 𝔮 and 𝔮̄ are chosen such that 𝔑 ⊗ Z_{q} = 𝔮̄^{n}. We may assume

ξ′_{q}(a, b) = ( \begin{matrix}a & 0 \\ 0 & b \end{matrix} ) .

Then \overline{η′_{q}} = (η_{1}, η_{2}) where η_{1} and η_{2}ε_{f,q}^{−1} are both unramified. Set 𝔧_{q} := ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) . Then

ϑ((a, b)𝔧_{q}) = I_{Z_{q}}(a)(I_{q^{n}Z_{q}} − \frac{1}{q}I_{q^{n−1}Z_{q}})(b),

and

ς(a, b) = \{ \begin{matrix}I_{Z_{q}}(a)I_{Z_{q}}(b), if q ∤ N_{ε_{f}}; \\ I_{Z_{q}}(a)I_{Z_{q}^{×}}(b)ε_{f,q}(b), if q | N_{ε_{f}} . \end{matrix}

Now

GL_{2}(Z_{q}) = Γ_{q}(1) ⨆ ⨆_{z=0}^{q−1} U(z)wΓ_{q}(1),

and

Γ_{q}(1) = ⨆_{y∈qZ_{q}/q^{n}Z_{q}} V (y)Γ_{q}(n).

so that

GL_{2}(Z_{q}) = ⨆_{y∈qZ_{q}/q^{n}Z_{q}} V (y)Γ_{q}(n) ⨆ ⨆_{z∈Z_{q}/qZ_{q}^{y∈qZq /q^{n}Zq}} U(z)wV (y)Γ_{q}(n).

Now V (y) = −wU(−y)w and wV (y) = U(−y)w. Thus

r_{ψ}(w, 1)ϑ((a, b)𝔧_{q}) = ϑ̂((a, b)𝔧_{q}) = \frac{1}{q^{n}} I_{q^{−n}Z_{q}^{×}}(a)I_{Z_{q}}(b),
r_{ψ}(U(−y), 1)ϑ̂((a, b)𝔧_{q}) = \frac{1}{q^{n}} ψ_{q}(yab)I_{q^{−n}Z_{q}^{×}}(a)I_{Z_{q}}(b),
r_{ψ}(wV (y), 1)ϑ(0) = r_{ψ}(U(−y)w, 1)ϑ(0) = r_{ψ}(U(−y), 1)ϑ̂(0) = ϑ̂(0) = 0,

and

r_{ψ}(V (y), 1)ϑ(0) = r_{ψ}(−wU(−y)w, 1)ϑ(0) = ∫ \frac{1}{q^{n}} ψ_{q}(yab)I_{q^{−n}Z_{q}^{×}}(a)I_{Z_{q}}(b)dadb
= \frac{1}{q^{n}} ∫ I_{Z_{q}}(ya)I_{q^{−n}Z_{q}^{×}}(a)da
= \{ \begin{matrix}0, if y ∉ q^{n}Z_{q}; \\ 1 − \frac{1}{q}, if y ∈ q^{n}Z_{q}. \end{matrix}

Thus

I = (1 − \frac{1}{q}) vol(Γ_{q}(n)) ∫ W_{F} (d(a))W_{Θ,ς}(d(a))|a|^{s−1}d^{×}a.

Now suppose first that q ∤ N_{ε_{f}}. Then η_{1} and η_{2} are both unramified and

W_{Θ,ς}(d(a)) = |a|^{1/2} \frac{η_{1}(aq) − η_{2}(aq)}{η_{1}(q) − η_{2}(q)}I_{Z_{q}}(a).

In this case π_{f,q} is either supercuspidal or ramified principal series ≃ π(μ_{1}, μ_{2}) with both μ_{1} and μ_{2} ramified. In any case, W_{F} (d(a)) = I_{Z_{q}^{×}}(a) and

I = (1 − \frac{1}{q}) vol(Γ_{q}(n)) = \frac{1}{q^{n−1}(q + 1)} · L_{q}(π̄_{f}, πη̄′, s)L_{q}(2s, ε_{K})^{−1}|_{s=1/2}.

Next suppose that q | N_{ε_{f}}. Then

W_{Θ,ς}(d(a)) = |a|^{1/2}η_{2}(a)I_{Z_{q}}(a).

As for W_{F}, we have

W_{F} (d(a)) = \{ \begin{matrix}ε_{f,q}^{−1}(a)I_{Z_{q}^{×}}(a), if q ∉ Σ′; \\ μ_{2}^{−1}(a)|a|^{1/2}I_{Z_{q}}(a), if q ∈ Σ′ and π_{f,q} ≃ π(μ_{1}, μ_{2}) with μ_{2} ramified. \end{matrix}

From this we find

I = \{ \begin{matrix}(1 − \frac{1}{q}) vol(Γ_{q}(n)) = \frac{1}{q^{n−1}(q+1)} · L_{q}(π̄_{f}, πη̄′, s)L_{q}(s, ε_{K})^{−1}|_{s=1/2}, if q ∈ Σ \ Σ′ \\ (1 − \frac{1}{q}) vol(Γ_{q}(n))(1 − μ_{2}^{−1}η_{2}(q)q^{−s}) = \frac{1}{q^{n−1}(q+1)} · \frac{L_{q}(π̄_{f} ,πη̄′ ,s)L_{q}(s,ε_{K})^{−1}}{1−μ_{1}^{−1}η_{1}(q)q^{−s}} |_{s=1/2}, if q ∈ Σ′. \end{matrix}

4.6.4. Case IV: q||N, q ∤ d_{K}. In this case, q is split in K i.e. q = 𝔮𝔮̄ and K ⊗Q_{q} ≃ Q_{q} ×Q_{q} corresponding to the completions at 𝔮 and 𝔮̄ respectively. We suppose that 𝔮 and 𝔮̄ are chosen such that 𝔑 ⊗ Z_{q} = 𝔮̄. We may assume

ξ′_{q}(a, b) = ( \begin{matrix}a & 0 \\ 0 & b \end{matrix} ) .

The character η̄′_{q} is identified with (η_{1}, η_{2}). Set 𝔧_{q} := ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) . Then

ϑ((a, b)𝔧_{q}) = I_{Z_{q}}(a)(I_{qZ_{q}} − \frac{1}{q}I_{Z_{q}})(b),

and

ς(a, b) = \{ \begin{matrix}I_{Z_{q}}(a)I_{Z_{q}}(b), if q ∤ N_{ε_{f}}; \\ I_{Z_{q}}(a)I_{Z_{q}^{×}}(b)ε_{f,q}(b), if q | N_{ε_{f}} . \end{matrix} .

I = \frac{1}{q + 1} (J(ς, ϑ) + qJ(w, ς̂, ϑ̂)) .

But ϑ(0) = 1 − \frac{1}{q} and ϑ̂(0) = 0. Hence

I = \frac{1}{q + 1}J(ς, ϑ) = \frac{1}{q + 1} · (1 − \frac{1}{q}) · ∫ W_{F} (d(a))W_{Θ,ς}(d(a))|a|^{s−1}d^{×}a,

where

W_{Θ,ς}(d(a)) = |a|^{1/2} ∫_{Q_{q}^{×}} ς(t, at^{−1})η_{1}(at^{−1})η_{2}(t)d^{×}t.

Suppose q ∤ N_{ε_{f}}. Then η_{1} and η_{2} are unramified and

W_{Θ,ς}(d(a)) = |a|^{1/2} ( ∑_{r+s=v_{q}(a)} η_{1}(q)^{r}η_{2}(q)^{s}) I_{Z_{q}}(a).

In this case, π_{f,q} is a special representation St(μ) with μ unramified and W_{F} (d(a)) = |a|μ^{−1}(a)I_{Z_{q}}(a). Hence

I = \frac{1}{(q + 1)} \frac{1 − 1/q}{(1 − q^{−1/2}μ^{−1}(q)η_{1}(q)q^{−s})(1 − q^{−1/2}μ^{−1}(q)η_{2}(q)q^{−s})} = \frac{1}{q + 1}L_{q}(π̄_{f}, πη̄′, s)L_{q}(2s, ε_{K})^{−1}|_{s=1/2}.

Next suppose q | N_{ε_{f}}, so that η_{1} is unramified and η_{2} is ramified but η_{2}ε_{f,q}^{−1} is unramified. Then

W_{Θ,ς}(d(a)) = |a|^{1/2}η_{2}(a).

In this case, π_{f,q} is a ramified principal series representation π(μ_{1}, μ_{2}) with say μ_{1} unramified and μ_{2} ramified. Since W_{F} (d(a)) = |a|^{1/2}μ_{2}^{−1}(a)I_{Z_{q}}(a), we get

I = \frac{1}{(q + 1)} \frac{1 − 1/q}{1 − μ_{2}^{−1}(q)η_{2}(q)q^{−s}} = \frac{1}{q + 1}L_{q}(π̄_{f}, πη̄′, s)L_{q}(2s, ε_{K})^{−1}|_{s=1/2}.

4.6.5. Case V: q||N, and q | d_{K}. Then n = 1. Recall that we have assumed q odd in this case. Let ϖ_{q} ∈ K_{q} := K ⊗ Q_{q} be such that Π_{q} := ϖ_{q}^{2} is a uniformizer in Z_{q}. We may assume

ξ′_{q}(ϖ_{q}) = ( \begin{matrix}0 & 1 \\ Π_{q} & 0 \end{matrix} ) .

Set 𝔧_{q} := ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ). First we suppose we are in

Subcase Va: q ∤ N_{ε_{f}}, i.e., q ∈ S(f). Then φ_{q} = ς ⊗ ϑ where

ς(a + bϖ_{q}) = I_{Z_{q}}(a)I_{Z_{q}}(b), ϑ((c + dϖ_{q})𝔧_{q}) = I_{Z_{q}}(c)I_{Z_{q}}(d),

so that

ς̂(a + bϖ_{q}) = q^{−1/2}I_{Z_{q}}(a)I_{\frac{1}{q} Z_{q}}(b),  ϑ̂((c + dϖ_{q})𝔧_{q}) = q^{−1/2}I_{Z_{q}}(c)I_{\frac{1}{q} Z_{q}}(d),

I = \frac{1}{q + 1} (J(ς, ϑ) + qJ(w, ς̂, ϑ̂)) .

Let β_{q} denote the matrix ( \begin{matrix}1 & 0 \\ 0 & Π_{q}^{−1} \end{matrix} ). Then

r_{ψ} (β_{q}, ϖ_{q}^{−1}) ς(a + bϖ_{q}) = I_{Z_{q}}(a)I_{\frac{1}{q} Z_{q}}(b) = q^{1/2}ς̂(a + bϖ_{q})

and likewise r_{ψ} (β_{q}, ϖ_{q}^{−1}) ϑ = q^{1/2}ϑ̂. Thus

I = \frac{1}{q + 1} (J(ς, ϑ) + \overline{η′_{q}}(ϖ_{q})J(β_{q}^{−1}w, ς, ϑ)) .

But π_{f,q} is special, say ≃ St(μ), hence W_{F} (gβ_{q}^{−1}w) = μ(Π_{q})W_{F} (g). Hence

I = \frac{(1 − μ(Π_{q})\overline{η′_{q}}(ϖ_{q}))}{q + 1}J(ς, ϑ) = \frac{2}{q + 1}J(ς, ϑ),

on account of our assumption that ε_{q}(f, χ^{−1}) = +1 and [Tu2], Prop. 1.7. Since \overline{η′_{q}} is unramified in this case, we can write η′_{q} = η_{1} ∘ N_{K_{q}/Q_{q}} = η_{2} ∘ N_{K_{q}/Q_{q}} where η_{1} is an unramified character of Q_{q}^{×} and η_{2} = η_{1} · ε_{K,q}. Then W_{Θ,ς}(d(a)) = |a|^{1/2}(η_{1}(a) + η_{2}(a))I_{Z_{q}}(a). Since W_{F} (d(a)) = |a|μ^{−1}(a)I_{Z_{q}}(a), we find

I = \frac{2}{q + 1} · \frac{1}{(1 − q^{−1/2}μ^{−1}(q)η_{1}(q)q^{−s})} = \frac{2}{q + 1} · L_{q}(π̄_{f}, πη̄′, s)L_{q}(2s, ε_{K})^{−1}|_{s=1/2}.

Subcase Vb: q | N_{ε_{f}}. Then

φ_{q} = ∑_{i≠j^{i,j∈Zq /qZq}} ε_{f,q}(i − j) · ς_{i} ⊗ ϑ_{j}

where

ς_{i}(a + bϖ_{q}) = I_{qZ_{q}+i}(a)I_{Z_{q}}(b), ϑ_{j}((c + dϖ_{q})𝔧_{q}) = I_{qZ_{q}+j}(c)I_{Z_{q}}(d),

I(φ_{q}) = \frac{1}{q + 1} · ∑_{i≠j^{i,j∈Zq /qZq}} ε_{f,q}(i − j) (J(ς_{i}, ϑ_{j}) + qJ(ς̂_{i}, ϑ̂_{j})) .

Note that ϑ̂_{j} is independent of j. Thus, for any fixed i, the sum ∑_{j≠i} ε_{f,q}(i − j)J(ς̂_{i}, ϑ̂_{j}) = 0. Also ϑ_{j}(0) = δ_{j0}. Consequently,

I(φ_{q}) = \frac{1}{q + 1} ∑_{i≠0} ε_{f,q}(i)J(ς_{i}, ϑ_{0}) = \frac{1}{q + 1}J(ς, ϑ_{0}),

where ς := ∑_{i≠0} ε_{f,q}(i)ς_{i}. Now W_{Θ,ς}(d(a)) = ε_{f,q}(a)(1 + ε_{K,q}(a))I_{Z_{q}^{×}}(a). Since π_{f,q} is ramified principal series of the form π(μ_{1}, μ_{2}) with μ_{1} unramified and μ_{2} ramified, we have W_{F} (d(a)) = |a|^{1/2}μ_{2}^{−1}(a)I_{Z_{q}}(a) and

I = \frac{1}{q + 1} = \frac{1}{q + 1} · L_{q}(π̄_{f}, πη̄′, s)L_{q}(2s, ε_{K})^{−1}|_{s=1/2}.

4.6.6. Case VI: q | d_{K}, q ∤ N. Again we may assume

ξ′_{q}(ϖ_{q}) = ( \begin{matrix}0 & 1 \\ π_{q} & 0 \end{matrix} ) .

Set 𝔧_{q} := ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ). Then

φ_{q} = ∑_{i=0}^{q−1} ς_{i} ⊗ ϑ_{i},

where

ς_{i}(a + bϖ_{q}) = I_{Z_{q}}(a)I_{\frac{i}{q} +Z_{q}}(b), ϑ_{i}((a + bϖ_{q})𝔧_{q}) = I_{Z_{q}}(a)I_{\frac{i}{q} +Z_{q}}(b).

Since q ∤ N, we have I = ∑_{i} J(ς_{i}, ϑ_{i}) = J(ς_{0}, ϑ_{0}). Since \overline{η′_{q}} is unramified in this case, we can write η′_{q} = η_{1}∘N_{K_{q}/Q_{q}} = η_{2}∘N_{K_{q}/Q_{q}} where η_{1} is an unramified character of Q_{q}^{×} and η_{2} = η_{1}·ε_{K,q}. Then W_{Θ,ς_{0}}(d(a)) =

|a|^{1/2}(η_{1}(a) + η_{2}(a))I_{Z_{q}}(a). If π_{f,q} ≃ π(μ_{1}, μ_{2}), then W_{F} (d(a)) = |a|^{1/2} \frac{μ^{−1}(aq)−μ_{2}^{−1}(aq)}{μ_{1}^{−1}(q)−μ_{2}^{−1}(q)}I_{Z_{q}}(a) and

I = \frac{1}{(1 − μ_{1}^{−1}(q)η_{1}(q)q^{−s})(1 − μ_{2}^{−1}(q)η_{1}(q)q^{−s})} = L_{q}(π̄_{f}, πη̄′, s)L_{q}(2s, ε_{K})^{−1}.

4.7. The explicit form of Waldspurger’s formula. We can now state the main result on the absolute value squared of the period integral L_{η,ξ}(F^{j}) defined in equation (4.3.6). We will need the class number formula L(1, ε_{K}) = 2πh_{K}/w_{K} \sqrt{|d_{K}|} and the volume of U′^{(1)}:

vol(U′^{(1)}) = ζ(2)^{−1} · ∏_{q^{nq} ||N} \frac{1}{q^{n_{q}−1}(q + 1)}.

Combining these with Cor. 4.26, equation (4.5.4) (with φ replaced by φ′) and the computations of the previous section, we obtain:

Theorem 4.28. Suppose cd_{K} is odd and η is a character of K of infinity type (−ℓ, 0) (ℓ = k + 2j) and finite type (c, 𝔑, ε_{f}^{−1}). Then

|L_{η′,ξ}(F^{j})|^{2} = C · L(\frac{1}{2}, \overline{π}_{f} × πη̄′) = C · L(\frac{1}{2}, π_{f} × π_{η′}),

with

(4.7.1)  C = \frac{Γ(j + 1)Γ(k + j)w_{K} \sqrt{|d_{K}|}ℑ(τ)^{−ℓ}}{(4π)^{k+2j+1} · h_{K}^{2} · c} · 2^{#S(f)} · ∏_{q|c} ζ_{K,q}(1).

Since L(\frac{1}{2}, π_{f} × π_{η′}) = L(f, χ^{−1}, 0), |Λ_{τ} |^{2}ℑ(τ) = vol(O_{c}) and h_{c}/h_{K} = c ∏_{q|c}(1 − ε_{K}(q)/q), we obtain Theorem 4.6 by combining Theorem 4.28 and Proposition 4.13.

5. Anticyclotomic p-adic L-functions

5.1. Periods and algebraicity. We will now use Theorem 4.6 of Section 4.1 to deduce algebraicity properties of the central critical values L(f, χ^{−1}, 0) attached to characters χ ∈ Σ_{cc}^{(2)}(𝔑). In order to do this, recall the dictionary between pairs (L, t) as in Section 4.1 and triples (E, t, ω) consisting of an elliptic curve over C, a point t on E of order N, and a differential ω ∈ Ω_{E/C}^{1}. Under this correspondence, the pair (L, t) corresponds to the triple (C/L, t, 2πidw), where the differential 2πidw arises from the standard coordinate w on C; in the other direction, the triple (E, t, ω) corresponds to the pair (Λ_{ω}, t) where 2πiΛ_{ω} is the period lattice attached to the differential ω. Viewing a nearly holomorphic modular form of weight k + 2j as a function on triples, we can rewrite the expression δ_{k}^{j}f(𝔞^{−1}, t) that appears in Theorem 4.6 as

δ_{k}^{j}f(𝔞^{−1}, t) = δ_{k}^{j}f(C/𝔞^{−1}, t, 2πidw) = δ_{k}^{j}f(𝔞 ∗ (A_{0}, t, 2πidw)),

where A_{0} := C/O_{c}, and we recall that the action of O_{c}-ideals of norm prime to N on marked elliptic curves with Γ-level structure of the form (A_{0}, t_{0}, ω_{0}) is the one described in equation (1.4.8) of Section 1.4.

Recall the triple (A, t_{A}, ω_{A}) with End_{F} (A) = O_{K} that was fixed until now. The curve A_{0} is the image of A by an isogeny φ_{0} : A⟶A_{0} of degree c. Let (A_{0}, t_{0}, ω_{0}) be the marked elliptic curve induced from (A, t_{A}, ω_{A}) via φ_{0}, i.e., the unique triple for which

(5.1.1)  φ_{0} : (A, t_{A}, ω_{A})⟶(A_{0}, t_{0}, ω_{0})

is an isogeny of marked elliptic curves with Γ-level structure in the sense of Definition 1.10. Given a Hecke character χ ∈ Σ_{cc}^{(2)}(𝔑) of infinity type (k + j, −j), it will be convenient to set

χ_{j} := χN^{j}

for the associated Hecke character of infinity type (k +2j, 0). Following the usual conventions, we will view χ_{j} as a multiplicative function on the fractional O_{c}-ideals that are prime to 𝔑c. This character satisfies

(5.1.2)  χ_{j}(x𝔞) = x^{k+2j}ε_{f}(x mod 𝔑)χ_{j}(𝔞),

for all x ∈ K^{×} that are prime to 𝔑c. After fixing the triple (A_{0}, t, 2πidw), with t an (arbitrarily chosen, but fixed from now on) generator of A_{0}[𝔑], the expression

χ_{j}^{−1}(𝔞)δ_{k}^{j}f(𝔞 ∗ (A_{0}, t, 2πidw))

depends only on the class of 𝔞 in Pic(O_{c}). (Cf. Lemma 4.5.) We can now restate Theorem 4.6 of Section 4.1 as follows:

Theorem 5.1. Let f be a normalised eigenform in S_{k}(Γ_{0}(N), ε_{f}) and let χ ∈ Σ_{cc}^{(2)}(𝔑) be a Hecke character of K of infinity type (k + j, −j). Then

(5.1.3)  C(f, χ, c)L(f, χ^{−1}, 0) = | ∑_{[𝔞]∈Pic(O_{c})} χ_{j}^{−1}(𝔞) · δ_{k}^{j}f(𝔞 ∗ (A_{0}, t, 2πidw))|^{2} ,

where the sum is taken over a system of representatives of the elements of Pic(O_{c}) that are prime to 𝔑c, and the constant C(f, χ, c) is given in Theorem 4.6.

Note that the sum appearing in the right-hand side of (5.1.3) does depend on the choice of generator t of A_{0}[𝔑], but only up to multiplication by an N-th root of unity; in particular, its absolute value is independent of the choice of t that was made.

For the purposes of algebraicity statements, p-adic interpolation, and the applications that are given in [BDP-cm] and [BDP-ch], it will be useful to have a formula in which the absolute value signs that occur in Theorem 5.1 are replaced by squares. In order to do this, we will need to examine the behavior of

(5.1.4)  J(f, χ) := ∑_{[𝔞]∈Pic(O_{c})} χ_{j}^{−1}(𝔞) · δ_{k}^{j}f(𝔞 ∗ (A_{0}, t, 2πidw))

under complex conjugation.

The choice of a primitive N-th root of unity ζ and of a square root of −N determines an Atkin-Lehner involution w_{N} acting on triples (E, t, ω) by the rule

w_{N}(E, t, ω) = (E/⟨t⟩, t′, \sqrt{−N}ω′),

where t′ is the image in E/⟨t⟩ of any element t′′ ∈ E[N] satisfying

⟨t, t′′⟩ = ζ

for the Weil pairing ⟨ , ⟩, and ω′ is the differential on E′ = E/⟨t⟩ which pulls back to ω under the natural projection. It is straightforward to verify that the function w_{N} is an involution on triples, and that it satisfies the commutation relation

(5.1.5)  𝔞 ∗ w_{N}(A_{0}, t, 2πidw) = w_{N}𝔞 ∗ (A_{0}, N𝔞^{−1}t, 2πidw).

Recall the decomposition N = 𝔑𝔑̄ of N as a product of two cyclic ideals of O_{c} of norm N. Choose an integral O_{c}-ideal 𝔟 and a nonzero element b_{N} ∈ O_{c} satisfying

(5.1.6)  (𝔟, Nc) = 1,  𝔟𝔑 = (b_{N}).

The multiplication by b_{N} map identifies the quotient A_{0}[N]/A_{0}[𝔑] with the submodule A_{0}[𝔑̄] of A_{0}[N]. Furthermore, the elliptic curve A_{0} and its differential dw are defined over R. Hence complex conjugation preserves them, but interchanges A_{0}[𝔑] and A_{0}[𝔑̄]. The pair (𝔟, b_{N}) therefore determines an element t′′ of A_{0}[N] satisfying

(5.1.7)  A_{0}[N] = (Z/NZ)t + (Z/NZ)t′′,  b_{N}t′′ = t̄.

This element is uniquely determined by b_{N} up to addition of a multiple of t. Therefore the primitive Nth root of unity

(5.1.8)  ζ := ⟨t, t′′⟩

depends only on b_{N} and not on the choice of t′′ satisfying (5.1.7). Let w_{N} denote the Atkin-Lehner involution associated to the root of unity ζ. If f is a modular form in S_{k}(Γ_{0}(N), ε_{f}), recall that f_{ρ} is the form in S_{k}(Γ_{0}(N), ε̄_{f}) whose fourier coefficients are the complex conjugates of those of f. If f is a normalised eigenform and a_{n} denotes the eigenvalue of the Hecke operator T_{n} acting on f, then we have the relation

(5.1.9)  ā_{n} = ε_{f}^{−1}(n)a_{n},

for all n which are relatively prime to N. In particular, the form f_{ρ} is also a normalised eigenform and corresponds to the twist of f by the character ε_{f}^{−1}. The following lemma is well-known.

Lemma 5.2. Suppose that f ∈ S_{k}(Γ_{0}(N), ε_{f}) is a newform. Then there exists a complex scalar w_{f} of norm one satisfying (for all triples (E, t, ω))

f_{ρ}(w_{N}(E, t, ω)) = w_{f}f(E, t, ω).

Proof. The operator w_{N} satisfies the following commutation relation relative to the Hecke operators:

(5.1.10)  T_{n}w_{N} = ⟨n⟩w_{N}T_{n},  ⟨n⟩w_{N} = w_{N}⟨n^{−1}⟩.

Equations (5.1.9) and (5.1.10) imply that the eigenvalue of T_{n} acting on w_{N}f_{ρ} is equal to a_{n}. By multiplicity one, it follows that w_{N}f_{ρ} is a non-zero scalar multiple of f, i.e., w_{N}f_{ρ} = w_{f}f, for some w_{f} ∈ C^{×}. The fact that w_{N} is defined over R, and hence commutes with the action of complex conjugation, implies also that w_{N}f = w̄_{f}f_{ρ}, and therefore |w_{f}|^{2} = 1 since w_{N}^{2} = 1. □

It should be noted that the scalar w_{f} is not entirely intrinsic to f, but depends on the choice of N-th root of unity ζ that was made in (5.1.8) prior to defining the Atkin-Lehner involution w_{N}. Over C, it is customary to take ζ = e^{\frac{2πi}{N}} but our choice of ζ may differ.

After these preliminaries, we define a complex scalar of norm one by the rule:

(5.1.11)  w(f, χ) := w_{f} · ε_{f}(N𝔟)^{−1}χ_{j}(𝔟)(−N)^{k/2+j}b_{N}^{−k−2j}.

Ostensibly, this scalar depends on the choice of (𝔟, b_{N}) satisfying (5.1.6), but in fact we have:

Lemma 5.3. The scalar w(f, χ) satisfies the following properties:

(1) It depends only on f and χ and not on the choice of pair (𝔟, b_{N}) satisfying (5.1.6);

(2) It belongs to the finite extension L of K generated by K_{f}, K_{χ}, and \sqrt{−N};

(3) For all σ ∈ Gal(L/K),

w(f^{σ}, χ^{σ}) = w(f, χ)^{σ}.

Proof. Properties (2) and (3) follow directly from the definition of w(f, χ). The truth of (1) follows from Theorem 5.4 below (since none of the terms other than w(f, χ) that appear in (5.1.12) depend on (𝔟, b_{N})) but it may be helpful to supply an independent, self-contained argument. If the pair (𝔟, b_{N}) is replaced by the pair (𝔟′, b′_{N}), then

𝔟′ = 𝔟(a), b′_{N} = b_{N}a,

where a is an element of K^{×} which is prime to 𝔑c. The conditions (5.1.7) and (5.1.8) that are required to be satisfied by b_{N} and b′_{N} imply that a ≡ 1 (mod 𝔑̄). The constants w(f, χ) attached to the choices (𝔟, b_{N}) and (𝔟′, b′_{N}) therefore differ by a factor of

ε_{f}(aā)^{−1}χ_{j}(a)a^{−k−2j} = ε_{f}(a mod 𝔑)^{−1}χ_{j}(a)a^{−k−2j}.

But this factor is equal to 1, by (5.1.2). □

Theorem 5.4. Let f be a normalised eigenform in S_{k}(Γ_{0}(N), ε_{f}) and let χ ∈ Σ_{cc}^{(2)}(𝔑) be a Hecke character of K of infinity type (k + j, −j). Then

(5.1.12)  C(f, χ, c)L(f, χ^{−1}, 0) = w(f, χ) ( ∑_{[𝔞]∈Pic(O_{c})} χ_{j}^{−1}(𝔞) · δ_{k}^{j}f(𝔞 ∗ (A_{0}, t, 2πidw)))^{2} ,

where the constants C(f, χ, c) and w(f, χ) are described in Theorem 4.6 and in equation (5.1.11) respectively.

Proof. Theorem 5.4 is proved by computing the effect of complex conjugation on the quantity J(f, χ) of equation (5.1.4). Observe that

(1) Since \overline{(A_{0}, 2πidw)} = (A_{0}, 2πidw) and b_{N} satisfies (5.1.7) and (5.1.8), the action of complex conjugation on (A_{0}, t, 2πidw) is given by

\overline{(A_{0}, t, 2πidw)} = (A_{0}, t̄, 2πidw) = 𝔟 ∗ w_{N}(A_{0}, t, b_{N} \sqrt{−N}^{−1}2πidw).

(2) The action of complex conjugation on χ_{j}^{−1}(𝔞) is given by

\overline{χ_{j}^{−1}(𝔞)} = ε_{f}(N𝔞)χ_{j}^{−1}(𝔞̄).

Hence we have

\overline{χ_{j}^{−1}(𝔞)δ_{k}^{j}f(𝔞 ∗ (A_{0}, t, 2πidw))} = ε_{f}(N𝔞)χ_{j}^{−1}(𝔞̄)δ_{k}^{j}f_{ρ}(𝔞̄ ∗ (A_{0}, t̄, 2πidw))

= ε_{f}(N𝔞)χ_{j}^{−1}(𝔞̄)δ_{k}^{j}f_{ρ}(𝔞̄𝔟 ∗ w_{N}(A_{0}, t, b_{N} \sqrt{−N}^{−1}2πidw)) (5.1.13)

= (−N)^{k/2+j}b_{N}^{−k−2j}ε_{f}(N𝔞)χ_{j}^{−1}(𝔞̄) · δ_{k}^{j}f_{ρ}(𝔞̄𝔟 ∗ w_{N}(A_{0}, t, 2πidw)).

But now, by (5.1.5), we have:

δ_{k}^{j}f_{ρ}(𝔞̄𝔟 ∗ w_{N}(A_{0}, t, 2πidw)) = δ_{k}^{j}f_{ρ}(w_{N} 𝔞̄𝔟 ∗ (A_{0}, (N𝔞̄𝔟)^{−1}t, 2πidw))
(5.1.14)  = w_{f}ε_{f}(N𝔞̄𝔟)^{−1} · δ_{k}^{j}f(𝔞̄𝔟 ∗ (A_{0}, t, 2πidw)).

Combining equations (5.1.13) and (5.1.14), we obtain

\overline{χ_{j}^{−1}(𝔞)δ_{k}^{j}f(𝔞 ∗ (A_{0}, t, 2πidw))}
= w_{f} · (−N)^{k/2+j}b_{N}^{−k−2j}χ_{j}(𝔟)ε_{f}(N𝔟)^{−1}χ_{j}(𝔞̄𝔟)^{−1}δ_{k}^{j}f(𝔞̄𝔟 ∗ (A_{0}, t, 2πidw)).

Summing this relation over all classes 𝔞 ∈ Pic O_{c}, we obtain

\overline{J(f, χ)} = w(f, χ)J(f, χ),

and Theorem 5.4 follows. □

We now turn to the algebraicity properties of L(f, χ^{−1}, 0). We begin by defining a complex period attached to K. For this, we observe that the complex elliptic curve A_{0} has endomorphism ring equal to the order O_{c} of conductor c, and therefore is defined over a subfield H_{c} of C which is isomorphic to the ring class field of K of conductor c. The choice of the differential ω_{0} ∈ Ω^{1}(A_{0}/H_{c}) determined by (5.1.1) determines a complex period Ω, defined as the non-zero complex scalar satisfying

(5.1.15)  ω_{0} = Ω · 2πidw,

where w is the standard complex coordinate on A_{0}(C) = C/O_{c}.

Theorem 5.5 below asserts that the ratios w^{−1}(f, χ)C(f, χ, c)L(f, χ^{−1}, 0)/Ω^{2(k+2j)} are algebraic numbers. In order to make a more precise claim about the fields of definition, we remark that the point t_{0} belongs (by assumption) to the 𝔑-torsion subgroup of A_{0}, which is defined over H_{c}. Let H′_{c} be the abelian extension of H_{c} over which the individual 𝔑-torsion points of A_{0} are defined, so that in particular the pair (A_{0}, t_{0}) is defined over H′_{c}. The Galois group of Gal(H′_{c}/H_{c}) is canonically identified with a subgroup of (Z/NZ)^{×} via its faithful action on A_{0}[𝔑]. Let H̃_{c} ⊂ H′_{c} be the subfield which is fixed by ker(ε_{f}). Let F ⊂ C be the finite extension of K generated by H̃_{c}, by the values of the Hecke character χ on 𝔸_{K,f}^{×}, and by the Fourier coefficients of f. We can now state Shimura’s algebraicity theorem on the special values L(f, χ^{−1}, 0) in a precise form.

Theorem 5.5. For all χ ∈ Σ_{cc}^{(2)}(𝔑) of infinity type (k + j, −j), the quantity

L_{alg}(f, χ^{−1}, 0) := w(f, χ)^{−1}C(f, χ, c) · L(f, χ^{−1}, 0)/Ω^{2(k+2j)}

belongs to F.

Proof. By Theorem 5.4,

w(f, χ)^{−1}C(f, χ, c)L(f, χ^{−1}, 0) = ( ∑_{[𝔞]∈Pic(O_{c})} χ_{j}^{−1}(𝔞) · δ_{k}^{j}f(𝔞 ∗ (A_{0}, t_{0}, 2πidw)))^{2}

= ( ∑_{[𝔞]∈Pic(O_{c})} χ_{j}^{−1}(𝔞) · δ_{k}^{j}f(𝔞 ∗ (A_{0}, t_{0}, Ω^{−1}ω_{0})))^{2}

= Ω^{2(k+2j)} ( ∑_{[𝔞]∈Pic(O_{c})} χ_{j}^{−1}(𝔞) · δ_{k}^{j}f(𝔞 ∗ (A_{0}, t_{0}, ω_{0})))^{2} .

It follows from Lemma 1.5 that

(5.1.16)  L_{alg}(f, χ^{−1}, 0) = ( ∑_{[𝔞]∈Pic(O_{c})} χ_{j}^{−1}(𝔞) · Θ_{Hodge}^{j}f(𝔞 ∗ (A_{0}, t_{0}, ω_{0})))^{2} .

Part 1 of Proposition 1.12 implies that the terms Θ_{Hodge}^{j}f(𝔞 ∗ (A_{0}, t_{0}, ω_{0})) belong to F. Theorem 5.5 follows. □

Remark 5.6. The datum of O_{c} determines the elliptic curve A_{0}/H_{c} together with the embedding of H_{c} into C. Both sides of (5.1.16) depend on the further choice of a regular differential ω_{0} on A_{0}/H_{c} which was determined by our choice of ω_{A}. Note that a change in ω_{A} (or ω_{0}) affects both sides of (5.1.16) in the same way.

5.2. p-adic interpolation. Let p be a rational prime which splits in K/Q, and fix a prime 𝔭 of K above p. Extending the associated embedding of K into Q_{p} to an embedding ι_{p} : F ⟶C_{p}. The special values L_{alg}(f, χ^{−1}, 0) can be viewed, through the embedding ι_{p}, as p-adic numbers. The following theorem gives a p-adic formula for these special values, in terms of the Atkin-Serre operator θ on p-adic modular forms.

Theorem 5.7. For all χ ∈ Σ_{cc}^{(2)}(𝔑) of infinity type (k + j, −j),

L_{alg}(f, χ^{−1}, 0) = ( ∑_{𝔞∈Pic(O_{c})} χ_{j}^{−1}(𝔞)(θ^{j}f)(𝔞 ∗ (A_{0}, t_{0}, ω_{0})))^{2} .

Proof. The fact that p is split in K implies that the elliptic curve ι_{p}(A_{0}) has good ordinary reduction. By part 3 of Proposition 1.12, combined with (5.1.16), we have:

(5.2.1)  L_{alg}(f, χ^{−1}, 0) = ( ∑_{[𝔞]∈Pic(O_{c})} χ_{j}^{−1}(𝔞) · Θ_{Frob}^{j}f(𝔞 ∗ (A_{0}, t_{0}, ω_{0})))^{2} .

Theorem 5.7 now follows from Lemma 1.7. □

Although the set Σ_{cc}^{(2)}(𝔑) is infinite, its elements take values in a finite extension of K. By possibly enlarging the finite extension F of K that appears in the statement of Theorem 5.5, we will assume that it contains the values χ(𝔞) as χ ranges over all characters in Σ_{cc}^{(2)}(𝔑) and 𝔞 ranges over 𝔸_{K,f}^{×}.

Let 𝔸′_{K,f} denote the subgroup of 𝔸_{K,f}^{×} of idèles which are prime to p, and choose any prime 𝔭_{F} of F above 𝔭. We observe that the values χ(𝔞) as 𝔞 ranges over 𝔸′_{K,f} are integral at 𝔭_{F}, i.e., they belong to the

ring of integers O_{F,𝔭_{F}} of the completion F_{𝔭_{F}}. It follows that Σ_{cc}^{(2)}(𝔑) is naturally embedded in the space F(𝔸′_{K,f}, O_{F,𝔭_{F}}) of O_{F,𝔭_{F}} -valued functions on 𝔸′_{K,f}. We equip Σ_{cc}^{(2)}(𝔑) with the topology induced by the compact open topology on this function space, i.e., the topology of uniform convergence on 𝔸′_{K,f} relative

to the p-adic topology on O_{F,𝔭}. Let Σ̂_{cc}(𝔑) be the completion of Σ_{cc}^{(2)}(𝔑) relative to this topology.

To p-adically interpolate the values L_{alg}(f, χ^{−1}, 0) we need to modify them by dropping a suitable Euler factor at p, and multiplying by a suitable p-adic period. We begin by attaching to A_{0} a p-adic period Ω_{p} as follows. Let A_{0} be a good integral model of A_{0} over O_{C_{p}}. The formal completion Â_{0} of A_{0} along its identity section is (non-canonically) isomorphic to 𝔾̂_{m} over O_{C_{p}}; fix such an isomorphism ι : Â_{0}⟶𝔾̂_{m}. (This amounts to fixing an isomorphism between the p-divisible groups μ_{p^{∞}} and A_{0}[𝔭^{∞}], which is determined up to a scalar in Z_{p}^{×}.) Fixing the isomorphism ι once and for all, we define Ω_{p} ∈ C_{p}^{×} by the rule, analogous to (5.1.15)

(5.2.2)  ω_{0} = Ω_{p} · ω_{can}, where ω_{can} := ι^{∗} \frac{du}{u},

and u denotes the standard coordinate on 𝔾̂_{m}.

For all χ ∈ Σ_{cc}^{(2)}(𝔑) of infinity type (k + j, −j), we set

(5.2.3)  L_{p}(f, χ) : = Ω_{p}^{2(k+2j)}(1 − χ^{−1}(𝔭̄)a_{p} + χ^{−2}(𝔭̄)ε_{f}(p)p^{k−1})^{2}L_{alg}(f, χ^{−1}, 0)
(5.2.4)  = Ω_{p}^{2(k+2j)}(1 − α_{p}χ^{−1}(𝔭̄))^{2}(1 − β_{p}χ^{−1}(𝔭̄))^{2}L_{alg}(f, χ^{−1}, 0),

where α_{p}, β_{p} denote the parameters of f at p described at the beginning of Section 4.1.

Remark 5.8. Note that both L_{alg}(f, χ) and Ω_{p} depend on the choice of the differential ω_{A} on A, but that the ratio L_{alg}(f, χ)/Ω_{p}^{2(k+2j)} does not depend on this choice, once an isomorphism ι between Â_{0} and Ĝ_{m} has been chosen. Replacing ι by a Z_{p}^{×}–multiple aι has the effect of multiplying L_{p}(f, χ) by a^{2(k+2j)}.

Recall the form f^{♭} = f|_{(V U−UV)} that was introduced in equation (3.8.4).

Theorem 5.9. Assume that p is split in K/Q. For all χ ∈ Σ_{cc}^{(2)}(𝔑) of infinity type (k + j, −j) (with j ≥ 0), we have

L_{p}(f, χ) = ( ∑_{[𝔞]∈Pic(O_{c})} χ_{j}^{−1}(𝔞) · θ^{j}f^{♭}(𝔞 ∗ (A_{0}, t, ω_{can})))^{2} .

Proof. Set

S_{χ} := ∑_{[𝔞]} χ_{j}^{−1}(𝔞) · θ^{j}f(𝔞 ∗ (A_{0}, t_{0}, ω_{0})),

and

S_{χ}^{♭} := ∑_{[𝔞]} χ_{j}^{−1}(𝔞) · θ^{j}f^{♭}(𝔞 ∗ (A_{0}, t_{0}, ω_{0})).

Now p^{j}a_{p} · θ^{j}f = θ^{j}f|T_{p} = θ^{j}f|(U + ε_{f}(p)p^{k+2j−1}V) and

(θ^{j}f|V)(𝔞 ∗ (A_{0}, t_{0}, ω_{0})) = (θ^{j}f)(𝔭̄^{−1}𝔞 ∗ (A_{0}, t_{0}, ω_{0})).

Thus

θ^{j}f^{♭}(𝔞 ∗ (A_{0}, t_{0}, ω_{0})) = {θ^{j}f|(V U − UV)}(𝔞 ∗ (A_{0}, t_{0}, ω_{0}))
= {θ^{j}f|(1 − T_{p}V + ε_{f}(p)p^{k+2j−1}V^{2})}(𝔞 ∗ (A_{0}, t_{0}, ω_{0}))
= θ^{j}f(𝔞 ∗ (A_{0}, t_{0}, ω_{0})) − p^{j}a_{p} · θ^{j}f(𝔭̄^{−1}𝔞 ∗ (A_{0}, t_{0}, ω_{0})) +
ε_{f}(p)p^{k+2j−1}θ^{j}f(𝔭̄^{−2}𝔞 ∗ (A_{0}, t_{0}, ω_{0})).

Multiplying this equation by χ_{j}^{−1}(𝔞) and summing over all the classes [𝔞] ∈ Pic(O_{c}) gives the identity

S_{χ}^{♭} = {1 − a_{p}χ^{−1}(𝔭̄) + ε_{f}(p)p^{k−1}χ^{−1}(𝔭̄^{2})} S_{χ}.

The result now follows from Theorem 5.7 combined with the homogeneity properties of the p-adic modular form θ^{j}f^{♭} of weight k + 2j. □

Proposition 5.10. The function χ ↦ L_{p}(f, χ) extends to a continuous function on Σ̂_{cc}(𝔑).

Proof. Let χ_{1}, χ_{2} ∈ Σ_{cc}^{(2)}(𝔑) be two elements (of infinity type (k + j_{1}, −j_{1}) and (k + j_{2}, −j_{2}) respectively) satisfying

χ_{1}(𝔞) ≡ χ_{2}(𝔞) (mod 𝔭^{M}), for all 𝔞 ∈ 𝔸′_{K,f}.

By evaluating at idèles in 𝔸′_{K,f} that are congruent to 1 modulo 𝔑, we see that necessarily

j_{1} ≡ j_{2} (mod (p − 1)p^{M−1}).

Now we observe that, since

θ^{j}f^{♭}(Tate(q), t, ω_{can}) = ∑_{(p,n)=1} n^{j}a_{n}q^{n},

the q-expansions of θ^{j_{1}}f and θ^{j_{2}}f are congruent modulo p^{M}, and therefore agree modulo 𝔭^{M}. If E is any ordinary elliptic curve over O_{F 𝔭}, and ω_{can} is any canonical differential on it as in (5.2.2), it follows that

θ^{j_{1}}f^{♭}(E, t, ω_{can}) ≡ θ^{j_{2}}f^{♭}(E, t, ω_{can}) (mod 𝔭^{M}).

(Cf. for example Sec.I.3.5 of [Gou].) It follows from the formula for L_{p}(f, χ) given in Theorem 5.9 that

L_{p}(f, χ_{1}) ≡ L_{p}(f, χ_{2}) (mod 𝔭^{M}).

The proposition follows. □

The function L_{p}(f, ·) on Σ̂_{cc}(𝔑) is a type of anticyclotomic p-adic L-function attached to f and K (and the triple (c, 𝔑, ε_{f})).

Remark 5.11. The p-adic L-functions attached to Rankin convolutions of p-adic families of modular forms have been constructed in great generality by Hida [Hi1]. In fact, our p-adic L-function L_{p}(f, ·) is the restriction of a more general “two-variable p-adic L-function” defined over Σ̂(𝔑), the existence of which can be deduced from the main result of [Hi1].

Note that one obtains from Hida’s work two different p-adic L-functions by interpolating the L-values corresponding to critical characters in Σ^{(1)}(𝔑) and Σ^{(2)}(𝔑) respectively. The p-adic L-function obtained by interpolating L(f, χ^{−1}, 0) with χ ∈ Σ^{(1)}(𝔑) has received much attention in the literature; for instance, it is studied in the article [PR1] of Perrin-Riou (for k = 2) and in [Ne2] (for k even and ≥ 2.) Our focus in this article has been instead on the p-adic L-function obtained by p-adic interpolation of the special values corresponding to (central critical characters) χ ∈ Σ^{(2)}(𝔑).

5.3. The main theorem. For the convenience of the reader, we collect the notations and the running assumptions that were made in the previous sections and are in force in the statement of Theorem 5.13 below.

Assumption 5.12. (1) The form f is a normalised cuspidal eigenform in S_{k}(Γ_{0}(N), ε_{f}). (2) c is an odd rational integer prime to Nd_{K}.

(3) The quadratic imaginary field K has odd discriminant and satisfies the Heegner hypothesis stated in Assumption 1.9, so that the order O_{c} of K of conductor c admits a cyclic ideal 𝔑 of norm N.

(4) The sets Σ_{cc}^{(1)}(𝔑) and Σ_{cc}^{(2)}(𝔑) consist of characters χ of finite type (c, 𝔑, ε_{f}) and satisfying ε_{q}(f, χ^{−1}) = +1 for all finite primes q, as described in Defn. 4.4 and the subsequent paragraph.

(5) The rational prime (p) = 𝔭𝔭̄ is split in K/Q and prime to Nc.

A character χ ∈ Σ_{cc}^{(1)}(𝔑) can be approximated by elements of Σ_{cc}^{(2)}(𝔑) (relative to the topology on Σ_{cc}(𝔑) discussed in the previous section) as follows. Let h denote the class number of K, and let ψ_{t} be the Hecke character of K of infinity type (th, −th) and trivial central character defined by

ψ_{t}(𝔞) = a^{t}/ā^{t}, where (a) = 𝔞^{h}.

If t is a sufficiently large positive integer, then the Hecke character χψ_{t} belongs to Σ_{cc}^{(2)}(𝔑), and it converges to χ as t converges to 0 in Z/(p − 1)Z × Z_{p}. This fact allows us to view Σ_{cc}^{(1)}(𝔑) as a subset of Σ̂_{cc}(𝔑).

The following Theorem, which relates the value of L_{p}(f, χ) at χ ∈ Σ_{cc}^{(1)}(𝔑) (which lies outside the range of interpolation for the p-adic L-function) to Abel-Jacobi images of generalized Heegner cycles, is the main result of this paper.

Theorem 5.13. Suppose that χ ∈ Σ_{cc}^{(1)}(𝔑) is a character of infinity type (k − 1 − j, 1 + j), with 0 ≤ j ≤ r. Then

\frac{L_{p}(f, χ)}{Ω_{p}^{2(r−2j)}} = (1 − χ^{−1}(𝔭̄)a_{p} + χ^{−2}(𝔭̄)ε_{f}(p)p^{k−1})^{2} ×

(\frac{c^{−j}}{j!} ∑_{[𝔞]∈Pic(O_{c})} χ^{−1}(𝔞)N(𝔞) · AJ_{F} (Δ_{φ_{𝔞}φ_{0}})(ω_{f} ∧ ω_{A}^{j}η_{A}^{r−j}))^{2} .

Proof. The proof of Proposition 5.10 shows that the formula in Theorem 5.9 for L_{p}(f, χ) at χ ∈ Σ_{cc}^{(2)}(𝔑) extends to χ ∈ Σ_{cc}^{(1)}(𝔑) in the obvious way, and gives

L_{p}(f, χ) = ( ∑_{[𝔞]∈Pic(O_{c})} χ_{−1−j}^{−1}(𝔞) · θ^{−1−j}f^{♭}(𝔞 ∗ (A_{0}, t_{0}, ω_{can})))^{2} .

Therefore, by (5.2.2) and the fact that θ^{−1−j}f^{♭} is a p-adic modular form of weight r − 2j, we have

\frac{L_{p}(f, χ)}{Ω_{p}^{2(r−2j)}} = ( ∑_{[𝔞]∈Pic(O_{c})} χ_{−1−j}^{−1}(𝔞) · θ^{−1−j}f^{♭}(𝔞 ∗ (A_{0}, t_{0}, ω_{0})))^{2} .

By Proposition 3.24,

(5.3.1)  \frac{L_{p}(f, χ)}{Ω_{p}^{2(r−2j)}} = (\frac{1}{j!} ∑_{[𝔞]∈Pic(O_{c})} χ_{−1−j}^{−1}(𝔞) · G_{j}^{♭}(𝔞 ∗ (A_{0}, t_{0}, ω_{0})))^{2} .

In view of Proposition 3.24 and of the relation θ^{j}f|T_{p} = p^{j}a_{p} · θ^{j}f, for j ≥ 0, one sees by p-adic approximation that

T_{p}G_{j} = p^{−1−j}a_{p}G_{j}.

Then, by Lemma 3.23,

G_{j}^{♭}(𝔞 ∗ (A_{0}, t_{0}, ω_{0})) = G_{j}(𝔞 ∗ (A_{0}, t_{0}, ω_{0})) − \frac{ϵ_{f}(p)a_{p}}{p^{r−j+1}} G_{j}(𝔭𝔞 ∗ (A_{0}, t_{0}, ω_{0})) + \frac{ϵ_{f}(p)}{p^{r−2j+1}} G(𝔭^{2}𝔞 ∗ (A_{0}, t_{0}, ω_{0})).

Substituting this expression for G_{j}^{♭}(𝔞 ∗ (A_{0}, t_{0}, ω_{0}) into (5.3.1) and rewriting the second and the third summands by substituting 𝔞 for 𝔞𝔭 and 𝔞𝔭^{2} respectively, we obtain

(5.3.2)  \frac{L_{p}(f, χ)}{Ω_{p}^{2(r−2j)}} = (1 − \frac{χ_{−1−j}(𝔭)a_{p}ε_{f}(p)}{p^{r−j+1}} + \frac{χ_{−1−j}^{2}(𝔭)ε_{f}(p)}{p^{r−2j+1}})^{2} ×

(\frac{1}{j!} ∑_{[𝔞]∈Pic(O_{c})} χ_{−1−j}^{−1}(𝔞) · G_{j}(𝔞 ∗ (A_{0}, t_{0}, ω_{0})))^{2} .

Using the fact that

χ_{−1−j}(𝔭) = χ(𝔭)p^{−1−j} = ε_{f}(p)^{−1}p^{r+1−j}χ(𝔭̄)^{−1},

the Euler factor that appears in (5.3.2) can be rewritten as

E_{p}(f, χ) := (1 − χ^{−1}(𝔭̄)a_{p} + χ^{−2}(𝔭̄)ε_{f}(p)p^{k−1})^{2} .

Now, applying Lemma 3.22 to the isogeny

φ_{𝔞}φ_{0} : (A, t_{A}, ω_{A})⟶𝔞 ∗ (A_{0}, t_{0}, ω_{0})

of degree cN(𝔞), and using the fact that χ_{−1−j}^{−1}(𝔞) = χ^{−1}(𝔞)N(𝔞)^{1+j}, we find

\frac{L_{p}(f, χ)}{Ω_{p}^{2(r−2j)}} = E_{p}(f, χ) (\frac{c^{−j}}{j!} ∑_{[𝔞]∈Pic(O_{c})} χ^{−1}(𝔞)N(𝔞) · AJ_{F} (Δ_{φ_{𝔞}φ_{0}})(ω_{f} ∧ ω_{A}^{j}η_{A}^{r−j}))^{2} ,

as was to be shown. □

Appendix A. Kuga-Sato schemes by Brian Conrad

The aim of this appendix is to explain the relative version of Deligne’s method for constructing a smooth projective compactification of the fiber powers E^{k} of the universal elliptic curve E with “enough level-N structure” over an open modular curve Y over Z[1/N] (for applications in this paper with Y = Y_{1}(N)). This was originally developed in 1968 for applications over finite fields F of characteristic not dividing N (see [De2, Lemma 5.5]), and later found uses for X(N) over Z[1/N] (see [Schol2, 4.2.1]). For applications over such fields F (e.g., Q or finite fields) one can compactify E_{F} → Y_{F} over the associated smooth complete modular curve X_{F} by using the technique of minimal regular proper models of relative smooth proper curves over a Dedekind base (such as E_{F} → Y_{F} relative to the Dedekind base X_{F}), together with their relation to Néron models of elliptic curves, and then try to explicitly resolve singularities of fiber powers over X_{F} of that minimal regular proper model. Thus, when working over such a field F there is no need for the concept of a generalized elliptic curve (which was introduced only in 1972 in the work of Deligne and Rapoport [DR], building on Artin’s theory of algebraic spaces).

The viewpoint of minimal regular proper models is insufficient in the relative situation over Z[1/N] since now X is 2-dimensional rather than Dedekind. In such settings we use the proper flat universal generalized elliptic curve \overline{E} → X over Z[1/N] (for a modular curve X classifying rigid fiberwise ample level-N structures on generalized elliptic curves over Z[1/N]-schemes) as a compactification of E over Z[1/N]. Such \overline{E} are smooth over Z[1/N] (see Lemma A.2) but not smooth over X, so for k ≥ 2 the compactification \overline{E}^{k} of E^{k} is not smooth over Z[1/N] (as we will see explicitly below). In Scholl’s work with X(N) over Z[1/N] in [Schol2, 4.2.1], for each k ≥ 2 he used Deligne’s method to construct a smooth projective Z[1/N]-scheme equipped with a proper birational map onto the fiber power \overline{E}^{k} over X such that the map is an isomorphism over E^{k} and can be described étale-locally near the fibers over the cuspidal locus on X. The method is a series of successive blow-ups, organized in terms of the number of coordinates of a geometric point ξ = (ξ_{1}, …, ξ_{k}) ∈ \overline{E}^{k} for which ξ_{i} is singular in its geometric fiber for \overline{E} → X.

The hard part is to give an intrinsic description of what to blow-up at each step; once we have defined an intrinsic algorithm, we can carry out computations étale-locally to see that we reach a smooth Z[1/N]- scheme. These étale-local computations are sketched over Q in Scholl’s work (see [Schol2, 2.0.1–2.1.1]) but the details on how to carry it out over Z[1/N] are omitted there (and the intrinsic definitions of what the pieces correspond to in terms of \overline{E}^{k} is not given). Thus, at the request of the referee, in this appendix we explain the procedure in more detail over Z[1/N].

We shall axiomatize the calculation so that it applies to “all” modular curves (with enough étale level structure). The intrinsic nature of the method also makes it applicable to cases in which the modular curve only exists as a Deligne–Mumford stack (such as X_{0}(N) over Z[1/N] for any N ≥ 1), but we leave that generalization to the interested reader. The “étale” nature of the level structure (i.e., using N-torsion level structures over Z[1/N]-schemes) is essential to the method because only in such cases can certain deformation-theoretic problems with generalized elliptic curves be reduced to the case of a Tate curve with geometrically irreducible fibers; see [DR, III, 1.4.2; VII, 2.1].

Fix an integer N ≥ 1, and let X be a modular curve over Z[1/N] classifying a rigid fiberwise ample level-N structure on generalized elliptic curves over Z[1/N]-schemes (e.g., Γ_{1}(N)-structures with N ≥ 5, or full level-N structures with N ≥ 3). Here, by “rigid” we mean that generalized elliptic curves equipped with such a level structure admit no nontrivial automorphisms. The work of Deligne and Rapoport provides such modular curves X as smooth proper Z[1/N]-schemes with fibers of pure dimension 1, equipped with a universal generalized elliptic curve \overline{E} → X. (Even though such an X is initially built only as an algebraic space, it is a scheme. This can be seen in a couple of ways, perhaps the most concrete being that the j-map from X to P_{Z[1/N]}^{1} is quasi-finite, and any algebraic space that is separated and quasi-finite over a noetherian scheme is a scheme [K, II, 6.16].)

Remark A.1. For the reader who is interested in schemes being projective rather than just proper, we make some side remarks now (not to be used in what follows). The fiberwise ample level structure on \overline{E} over X defines a closed subgroup scheme of the open X-smooth locus \overline{E}^{sm} that is finite étale over X and so is closed in \overline{E} with ideal sheaf in O\overline{E} that is a line bundle on \overline{E} whose inverse is fiberwise ample over X. But a fiberwise ample line bundle on a proper finitely presented scheme over a base S is relatively ample over S [EGA, IV_{3}, 9.6.4], so the projectivity and flatness of X over Z[1/N] implies that \overline{E} is projective and flat over Z[1/N]. Likewise, the fiber powers \overline{E}^{k} over X are projective and flat over Z[1/N] for all k ≥ 1. In particular, any scheme obtained from \overline{E}^{k} by a composition of successive blow-ups is projective over Z[1/N]. This ensures that the Z[1/N]-smooth compactification of \overline{E}^{k} built below is projective over Z[1/N].

We now recall that for any generalized elliptic curve f : E → S over a scheme, Deligne and Rapoport introduced canonical closed subscheme structures S_{∞} ⊂ S and E^{sing} ⊂ E respectively supported at the set of s ∈ S such that E_{s} is not k(s)-smooth and at the set of ξ ∈ E at which the proper fppf map E → S is not smooth. Explicitly, E^{sing} is defined by the annihilator ideal of Ω_{E/S}^{2} (the first Fitting ideal of Ω_{E/S}^{1}), and S_{∞} is defined to be the scheme-theoretic image of E^{sing} in S. The formation of both of these commutes with any base change on S (though this has some hidden subtleties for S_{∞}; see [Con, 2.1.11, 2.1.12]). We call these closed subschemes the “loci of non-smoothness” in S and E for f. Their compatibility with base change on S enables us to compute completions along these loci via deformation theory.

Let X_{∞} ⊂ X be the locus of non-smoothness for the universal generalized elliptic curve \overline{E} → X. Computations with the deformation theory of generalized elliptic curves equipped with ample level-N structure over Z[1/N] show that X_{∞} is (finite) étale over Z[1/N] (see [DR, III, 1.2(iv); IV, 3.4(ii)]). The structure of \overline{E} around \overline{E}^{sing} can also be understood via deformation theory, leading to:

Lemma A.2. The scheme \overline{E} is smooth over Z[1/N].

Proof. The problem is to prove smoothness at non-smooth points ξ in fibers over points x ∈ X_{∞}, and since \overline{E} is fppf over Z[1/N] it suffices to work on geometric fibers over Spec(Z[1/N]). In other words, for an algebraically closed field F of characteristic not dividing N and the universal generalized elliptic curve \overline{E}_{F} → X_{F}, we want to prove that the surface \overline{E}_{F} is smooth at points ξ ∈ \overline{E}(F) that are non-smooth in the fiber over x ∈ X_{∞}(F). It is equivalent to prove the formal smoothness of O_{\overline{E}_{F} ,ξ}^{∧} over F. But

O_{\overline{E}_{F} ,ξ}^{∧} coincides with the completed local ring at ξ on the formal completion of \overline{E}_{F} → X_{F} along x. This

latter formal completion is the universal deformation of (\overline{E}_{F})_{x} equipped with its ample level-N structure, and O_{X_{F} ,x}^{∧} is its universal deformation ring. Since char(F) ∤ N, by [DR, III, 1.2(iv); VII, (1.1.1), 1.11, 2.1] there is an F -isomorphism between the universal deformation ring O_{X_{F} ,x}^{∧} and F [[q]] such that the completed local ring at ξ is F [[q]]-isomorphic to F [[q, u, v]]/(uv − q) = F [[u, v]]. □

Now we shall prove a general resolution result for generalized elliptic curves over a family of smooth curves:

Theorem A.3. Let S be a scheme, X → S a smooth map with all fibers of pure dimension 1, and f : \overline{E} → X a generalized elliptic curve such that:

(1) the locus of non-smoothness X_{∞} ⊂ X for f is étale over S,

(2) the scheme \overline{E} is S-smooth.

For each k ≥ 1, let \overline{E}^{k} denote the kth fiber power over X.

There exists a smooth S-scheme Z_{k} and a proper birational map Z_{k} → \overline{E}^{k} that is an isomorphism over E^{k}. The map Z_{k} → \overline{E}^{k} is a composition of finitely many blow-ups.

We emphasize that although \overline{E} is assumed to be S-smooth, in practice it is not X-smooth, so the closed subscheme \overline{E}^{sing} (which encodes non-smoothness over X) is generally not empty. The proof of the theorem consists of giving an explicit definition of the blow-up process. If k = 1 then we may take Z_{1} = \overline{E} by hypothesis (2), so we now assume k ≥ 2.

By hypothesis (1), the pair (X, X_{∞}) looks étale-locally like (A_{R}^{1}, 0). Thus, the étale-local structure of relative semi-stable curves [FK, III, 2.7] and the “homogeneity” of \overline{E} around \overline{E}^{sing} (via translation by \overline{E}^{sm}) implies that Zariski-locally over an affine open Spec R in S the pair (\overline{E}, \overline{E}^{sing}) has a common étale neighborhood with

(Spec(R[q, u, v]/(uv − q)), {q = u = v = 0})

(see the proof of [DR, II, 1.16]). Up to permutation of coordinates, a geometric point ξ = (ξ_{1}, …, ξ_{k}) ∈ \overline{E}^{k} that is non-smooth over S has ξ_{1}, …, ξ_{r} non-smooth in \overline{E} over X and ξ_{r+1}, …, ξ_{k} smooth in \overline{E} over X for some r ≥ 2 (the case r = 1 being ruled out by the hypothesis that \overline{E} is S-smooth). Thus, (\overline{E}^{k}, ξ) has a common étale neighborhood with the spectrum of

(A.0.3) R[q, X_{1}, Y_{1}, …, X_{r}, Y_{r}, T_{r+1}, …, T_{k}]/(X_{1}Y_{1} = ⋯ = X_{r}Y_{r} = q) ≃

R[X_{1}, Y_{1}, …, X_{r}, Y_{r}, T_{r+1}, …, T_{k}]/(X_{1}Y_{1} = ⋯ = X_{r}Y_{r}).

Of course, we have an analogous ring for any permutation of the ξ_{i}’s.

Let \overline{F}^{k} denote the k-fold fiber product of \overline{E} over X_{∞}. We define a stratification of \overline{F}^{k} ↪ \overline{E}^{k} by closed subschemes

\overline{F}^{k} = F_{k}^{k} ⊋ F_{k−1}^{k} ⊋ ⋯ ⊋ F_{0}^{k} ⊇ F_{−1}^{k} = ∅

where, for 0 ≤ r ≤ k, F_{r}^{k} ⊆ \overline{F}^{k} is the scheme-theoretic union of the closed subschemes defined by requiring at least k − r factors to lie in \overline{E}^{sing}. For example, working étale locally over \overline{E}, we see that F_{k−2}^{k} is

supported at precisely the closed non-smooth locus for the fppf map \overline{E}^{k} → S.

Define E^{k}⟨0⟩ = \overline{E}^{k} and F_{i}^{k}⟨0⟩ = F_{i}^{k} for 0 ≤ i ≤ k. For 1 ≤ r ≤ k − 1, we recursively define E^{k}⟨r⟩ = Bl_{F_{r−1}^{k}⟨r−1⟩}(E^{k}⟨r − 1⟩), and we let F_{i}^{k}⟨r⟩ be the proper transform in E^{k}⟨r⟩ of F_{i}^{k}⟨r − 1⟩ for r ≤ i ≤ k − 1. (Equivalently, F_{i}^{k}⟨r⟩ is the blow-up of F_{i}^{k}⟨r − 1⟩ along F_{r−1}^{k}⟨r − 1⟩.)

We claim several properties:

(i) E^{k}⟨r⟩ and all F_{i}^{k}⟨r⟩ are S-flat, (ii) F_{r}^{k}⟨r⟩ is contained in the closed locus where the S-flat E^{k}⟨r⟩ is non-smooth over S for all 0 ≤ r ≤ k − 2 (so the map E^{k}⟨k − 1⟩ → E^{k}⟨0⟩ = \overline{E}^{k} is an isomorphism over the S-smooth locus of \overline{E}^{k}, which contains \overline{E}^{k}),

(iii) E^{k}⟨k − 1⟩ is S-smooth, (iv) the formation of these blow-ups and strict transforms commutes with any base change on S (via

the evident base change morphisms).

To verify these claims we may work étale-locally over a non-smooth point of \overline{E}^{k} over affine open Spec R ⊂ S, which amounts to replacing \overline{E}^{k} with

Ẽ^{m}⟨0⟩ = R[X_{1}, Y_{1}, …, X_{m}, Y_{m}, T_{m+1}, …, T_{k}]/(X_{1}Y_{1} = ⋯ = X_{m}Y_{m}),

where 2 ≤ m ≤ k.

We define F̃_{i}^{m}⟨0⟩ to be the R-flat locus in Ẽ^{m}⟨0⟩ where m − i pairs (X_{j}, Y_{j}) vanish. Using inductive definitions analogous to those above, we define Ẽ^{m}⟨r⟩ and F̃_{i}^{m}⟨r⟩ (with r ≤ i ≤ m − 1) for 0 ≤ r ≤ m − 1. We can replace the above claims with analogues in this new setting, so we aim to prove:

• F̃_{i}^{m}⟨r⟩ and Ẽ^{m}⟨r⟩ are R-flat and their formation commutes with base change on R;

• F̃_{r}^{m}⟨r⟩ is contained in the closed non-smooth locus for Ẽ^{m}⟨r⟩ over R for all 0 ≤ r ≤ m − 2 (so the blow-up steps are always isomorphisms over the smooth locus of the previous stage);

• Ẽ^{m}⟨m − 1⟩ is R-smooth. This will clearly finish the proof. The T_{m+1}, …, T_{k} just get “carried along”, so they can (and will) now be dropped.

It is easy to see that Ẽ^{m}⟨1⟩ has an open cover by 2m-copies U_{j} of A^{1} ×Ẽ^{m−1}⟨0⟩ such that U_{j} ∩F̃_{i}^{m}⟨1⟩ = A^{1} × F̃_{i−1}^{m−1}⟨0⟩ for 1 ≤ i ≤ m − 1. Here, we define Ẽ^{1}⟨0⟩ = Spec R[X_{1}, Y_{1}]/(X_{1}Y_{1}) and F̃_{0}^{1} = (0, 0).

By induction on r for each m (with the case r = 0 always trivial and the case r = 1 just settled for all m) we see that for 0 ≤ r ≤ m − 2 there exists an open cover of Ẽ^{m}⟨r⟩ by copies V_{j} of A^{r} × Ẽ^{m−r}⟨0⟩ with V_{j} ∩ F̃_{i}^{m}⟨r⟩ = A^{r} × F̃_{i−r}^{m−r}⟨0⟩ for all r ≤ i ≤ m − 1. Thus, F̃_{r}^{m}⟨r⟩ is contained in F̃_{0}^{m−r}, which in turn is contained in the closed locus of non-smooth points in Ẽ^{m−r}⟨0⟩ over R since m−r ≥ 2. These Zariski-local descriptions yield the desired R-flatness and compatibility with base change on R.

Taking r = m − 2 at the end of the induction, Ẽ^{m}⟨m − 2⟩ is covered by open subschemes R-isomorphic to A^{m−2} × Ẽ^{2}⟨0⟩. Since

Ẽ^{2}⟨0⟩ = Spec R[X_{1}, Y_{1}, X_{2}, Y_{2}]/(X_{1}Y_{1} − X_{2}Y_{2})

with F̃_{0}^{2}⟨0⟩ equal to the origin over R, so it remains to note that the R-scheme Bl_{(0)}(Ẽ^{2}⟨0⟩) is covered by copies of A^{3}.

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