CHOW-HEEGNER POINTS ON CM ELLIPTIC CURVES AND VALUES OF p-ADIC L-FUNCTIONS

MASSIMO BERTOLINI HENRI DARMON KARTIK PRASANNA

Contents

Introduction 1 1. Hecke characters and periods 9 1.1. Algebraic Hecke characters 9 1.2. Abelian varieties associated to characters of type (1, 0) 10 1.3. Complex periods and special values of L-functions 13 1.4. p-adic periods 14 2. p-adic L-functions and Rubin’s formula 15 2.1. The Katz p-adic L-function 15 2.2. p-adic Rankin L-series 17 2.3. A p-adic Gross-Zagier formula 19 2.4. A factorisation of the p-adic Rankin L-series 21 2.5. Proof of Rubin’s Theorem 22 2.6. Elliptic curves with complex multiplication 27 2.7. A special case 29 3. Chow-Heegner points 30 3.1. Motives for rational and homological equivalence 30 3.2. The motive of a Hecke character 31 3.3. Deligne-Scholl motives 32 3.4. Modular parametrisations attached to CM forms 33 3.5. Generalised Heegner cycles and Chow-Heegner points 36 3.6. A special case 37 4. Chow-Heegner points over ℂ_{p} 38 4.1. The p-adic Abel-Jacobi map 38 4.2. Rationality of Chow-Heegner points over ℂ_{p} 39 4.3. A special case 39 5. Chow-Heegner points over ℂ 42 5.1. The complex Abel-Jacobi map 42 5.2. Numerical experiments 43 References 47

Introduction

The theory of Heegner points supplies one of the most fruitful approaches to the Birch and Swinnerton-Dyer conjecture, leading to the best results for elliptic curves of analytic rank one. In spite of attempts to broaden the scope of the Heegner point construction ([BDG], [Da2], [Tr],...), all provable, systematic constructions of algebraic points on elliptic curves still rely on parametrisations of elliptic curves by modular or Shimura curves. The primary goal of this article is to explore new constructions of rational points on elliptic curves and abelian varieties in which, loosely speaking, Heegner divisors are replaced by higher-dimensional algebraic cycles on certain modular varieties. In general, the algebraicity of the resulting points depends on the validity of ostensibly difficult cases of the Hodge or Tate conjectures. One of the main results of this article (Theorem 8 of the introduction) illustrates how these algebraicity statements can sometimes be obtained unconditionally by exploiting the connection between the relevant “generalised Heegner cycles” and values of certain p-adic L-series.

During the preparation of this article, KP was supported partially by NSF grants DMS-1015173 and DMS-0854900.

In the course of our study, we are also led to a new proof of the main theorem of [Ru] and to a generalisation thereof (see Theorem 2 below), relating values of the Katz two-variable p-adic L-function to formal group logarithms of rational points on CM elliptic curves.

Rubin’s Theorem. The main theorem in [Ru] (Cf. Corollary 10.3 of loc. cit.) concerns an elliptic curve A over ℚ with complex multiplication by the ring of integers of a quadratic imaginary field K. A classical result of Deuring identifies the Hasse-Weil L-series L(A, s) of A with the L-series L(ν_{A}, s) attached to a Hecke character ν_{A} of K of infinity type (1, 0). When p is a prime which splits in K and does not divide the conductor of A, the Hecke L-function L(ν_{A}, s) has a p-adic analogue known as the Katz twovariable L-function attached to K. It is a p-adic analytic function, denoted ν ↦ ℒ_{p}(ν), on the space of Hecke characters equipped with its natural p-adic analytic structure. Section 2.1 recalls the definition of this L-function: the values ℒ_{p}(ν) at Hecke characters of infinity type (1 + j_{1}, −j_{2}) with j_{1}, j_{2} ≥ 0 are defined by interpolation of the classical L-values L(ν^{−1}, 0). Letting ν^{∗} := ν ∘ c, where c denotes complex conjugation on the ideals of K, it is readily seen by comparing Euler factors that L(ν, s) = L(ν^{∗}, s). A similar equality need not hold in the p-adic setting, because the involution ν ↦ ν^{∗} corresponds to the map (j_{1}, j_{2}) ↦ (j_{2}, j_{1}) on weight space and therefore does not preserve the lower right quadrant of weights of Hecke characters that lie in the range of classical interpolation. Since ν_{A} lies in the domain of classical interpolation, the p-adic L-value ℒ_{p}(ν_{A}) is a simple multiple of L(ν_{A}^{−1}, 0) = L(A, 1). Suppose that it vanishes. (This implies, by the Birch and Swinnerton-Dyer conjecture, that A(ℚ) is infinite.) The value ℒ_{p}(ν_{A}^{∗}) is a second, a priori more mysterious p-adic avatar of the leading term of L(A, s) at s = 1. Rubin’s theorem gives a formula for this quantity:

Theorem 1 (Rubin). Let ν_{A} be a Hecke character of type (1, 0) attached to an elliptic curve A/ℚ with complex multiplication. Then there exists a global point P ∈ A(ℚ) such that

(1)  ℒ_{p}(ν_{A}^{∗}) = Ω_{p}(A)^{−1} log_{ω_{A}}(P)^{2} (mod ℚ^{×}),

where • Ω_{p}(A) is the p-adic period attached to A as in Section 1.3;

• ω_{A} ∈ Ω^{1}(A/ℚ) is a regular differential on A over ℚ, and log_{ω_{A}} : A(ℚ_{p})⟶ℚ_{p} denotes the p-adic

formal group logarithm with respect to ω_{A}. The point P is of infinite order if and only if L(A, s) has a simple zero at s = 1.

(For a more precise statement without the ℚ^{×} ambiguity, see [Ru].) Formula (1) is peculiar to the padic world and suggests that p-adic L-functions encode arithmetic information that is not readily apparent in their complex counterparts. Inspired by Rubin’s work, Perrin-Riou has formulated a p-adic Beilinson conjecture in [PR2] of which Theorem 1 should be a special case.

The proof of Theorem 1 given in [Ru] breaks up naturally into two parts:

(1) Rubin exploits the Euler system of elliptic units to construct a global cohomology class κ_{A} belonging to a pro-p Selmer group Sel_{p}(A/ℚ) attached to A. The close connection between elliptic units and the Katz L-function is then parlayed into the explicit evaluation of two natural p-adic invariants attached to κ_{A}: the p-adic formal group logarithm log_{A,p}(κ_{A}) and the cyclotomic p-adic height ⟨κ_{A}, κ_{A}⟩:

(2)  log_{A,p}(κ_{A}) = (1 − β_{p}^{−1})^{−1}ℒ_{p}(ν_{A}^{∗})Ω_{p}(A),
(3)  ⟨κ_{A}, κ_{A}⟩ = (1 − α_{p}^{−1})^{−2}ℒ′_{p}(ν_{A})ℒ_{p}(ν_{A}^{∗}),

where

• α_{p} and β_{p} denote the roots of the Hasse polynomial x^{2} − a_{p}(A)x + p, ordered in such a way that ord_{p}(α_{p}) = 0 and ord_{p}(β_{p}) = 1;

• the quantity ℒ′_{p}(ν_{A}) denotes the derivative of ℒ_{p} at ν_{A} in the direction of the cyclotomic character.

If ℒ′_{p}(ν_{A}) is non-zero, then an argument based on Perrin-Riou’s p-adic analogue of the Gross-Zagier formula and the work of Kolyvagin implies that Sel_{p}(A/ℚ) ⊗ ℚ is a one-dimensional ℚ_{p}-vector space with κ_{A} as a generator. (Cf. Thm. 8.1 and Cor. 8.3 of [Ru].) Equations (2) and (3) then make it possible to evaluate the ratio

(4)  \frac{log_{A,p}^{2}(κ)}{⟨κ, κ⟩} = \frac{(1 − β_{p}^{−1})^{−2}ℒ_{p}(ν_{A}^{∗})Ω_{p}(A)^{2}}{(1 − α_{p}^{−1})^{−2}ℒ′_{p}(ν_{A})} ,

a quantity which does not depend on the choice of generator κ of the ℚ_{p}-vector space Sel_{p}(A/ℚ)⊗ℚ.

(2) Independently of the construction of κ_{A}, the theory of Heegner points can be used to construct a canonical point P ∈ A(ℚ), which is of infinite order when ℒ′_{p}(ν_{A}) ≠ 0. Its image κ_{P} ∈ Sel_{p}(A/ℚ) under the connecting homomorphism of Kummer theory supplies us with a second generator for Sel_{p}(A/ℚ) ⊗ ℚ. Furthermore, the p-adic analogue of the Gross-Zagier formula proved by Perrin-Riou in [PR1] shows that

(5)  ⟨κ_{P} , κ_{P} ⟩ = ℒ′_{p}(ν_{A})Ω_{p}(A)^{−1} (mod ℚ^{×}).

Rubin obtains Theorem 1 by setting κ = κ_{P} in (4) and using (5) to eliminate the quantities involving ⟨κ_{P} , κ_{P} ⟩ and ℒ′_{p}(ν_{A}).

The reader will note the key role that is played in Rubin’s proof by both the Euler systems of elliptic units and of Heegner points. The new approach to Theorem 1 described in Chapter 2 relies solely on Heegner points, and requires neither elliptic units nor Perrin-Riou’s p-adic height calculations. Instead, the key ingredient in this approach is the p-adic variant of the Gross-Zagier formula arising from the results of [BDP] which is stated in Theorem 2.9. This formula expresses p-adic logarithms of Heegner points in terms of the special values of a p-adic Rankin L-function attached to a cusp form f and an imaginary quadratic field K, and may be of some independent interest insofar as it exhibits a strong analogy with Rubin’s formula but applies to arbitrary—not necessarily CM—elliptic curves over ℚ. When f is the theta series attached to a Hecke character of K, Theorem 1 follows from the factorisation of the associated p-adic Rankin L-function into a product of two Katz L-functions, a factorisation which is a simple manifestation of the Artin formalism for these p-adic L-series.

It is expected that the statement of Theorem 1 should generalise to the setting where ν_{A} is replaced by an algebraic Hecke character ν of infinity type (1, 0) of a quadratic imaginary field K (of arbitrary class number) satisfying

(6)  ν|_{𝔸_{ℚ}} = ε_{K} · N,

where ε_{K} denotes the quadratic Dirichlet character associated to K/ℚ and N : 𝔸_{ℚ}^{×}⟶ℝ^{×} is the adèlic norm character. Chapter 2 treats this more general setting, which is not yet covered in the literature, although the original methods of [Ru] would probably extend as well with only technical complications. Assumption (6) implies that the classical functional equation relates L(ν, s) to L(ν, 2 − s). Assume further that the sign w_{ν} in this functional equation satisfies

(7)  w_{ν} = −1,

so that L(ν, s) vanishes to odd order at s = 1. For less serious reasons, it will also be convenient to make two further technical assumptions. Firstly, we assume that

(8)  The discriminant −D of K is odd.

Secondly, we note that assumption (6) implies that \sqrt{−D} necessarily divides the conductor of ν, and we further restrict the setting by imposing the assumption that

(9)  The conductor of ν is exactly divisible by \sqrt{−D}.

The statement of Theorem 2 below requires some further notions which we now introduce. Let E_{ν} be the subfield of ℂ generated by the values of the Hecke character ν, and let T_{ν} be its ring of integers. A general construction which is recalled in Sections 1.2 and 2.6 attaches to ν an abelian variety B_{ν} over K of dimension [E_{ν} : K], equipped with inclusions

T_{ν} ⊂ End_{K}(B_{ν}), E_{ν} ⊂ End_{K}(B_{ν}) ⊗ ℚ.

Given λ ∈ T_{ν}, denote by [λ] the corresponding endomorphism of B_{ν}, and set

(10)  Ω^{1}(B_{ν}/E_{ν})^{T_{ν}} := {ω ∈ Ω^{1}(B_{ν}/E_{ν}) such that [λ]^{∗}ω = λω, ∀λ ∈ T_{ν}} ,
(11)  (B_{ν}(K) ⊗ E_{ν})^{T_{ν}} := {P ∈ B_{ν}(K) ⊗_{ℤ} E_{ν} such that [λ]P = λP, ∀λ ∈ T_{ν}} .

The vector space Ω^{1}(B_{ν}/Eν)^{T_{ν}} is one-dimensional over E_{ν}. The results of Gross-Zagier and Kolyvagin, which continue to hold in the setting of abelian variety quotients of modular curves, also imply that (B_{ν}(K) ⊗ E_{ν})^{T_{ν}} is one-dimensional over E_{ν} when L(ν, s) has a simple zero at s = 1.

After fixing a p-adic embedding K ⊂ ℚ_{p}, the formal group logarithm on B_{ν} gives rise to a bilinear pairing

⟨ , ⟩ : Ω^{1}(B_{ν}/K) × B_{ν}(K) ⟶ ℚ_{p}
(ω, P) ↦ log_{ω} P,

satisfying ⟨[λ]^{∗}ω, P ⟩ = ⟨ω, [λ]P ⟩ for all λ ∈ T_{ν}. This pairing can be extended by E_{ν}-linearity to an E_{ν} ⊗ ℚ_{p}-valued pairing between Ω^{1}(B_{ν}/E_{ν}) and B_{ν}(K) ⊗ E_{ν}. When ω and P belong to these E_{ν}-vector spaces, we will continue to write log_{ω}(P) for ⟨ω, P ⟩.

Theorem 2. Let ν be an algebraic Hecke character of infinity type (1, 0) satisfying (6), (7), (8) and (9) above. Then there exists P_{ν} ∈ B_{ν}(K) such that

ℒ_{p}(ν^{∗}) = Ω_{p}(ν^{∗})^{−1} log_{ω_{ν}} (P_{ν})^{2} (mod E_{ν}^{×}),

where Ω_{p}(ν^{∗}) ∈ ℂ_{p} is the p-adic period attached to ν in Definition 1.14, and ω_{ν} is a non-zero element of Ω^{1}(B_{ν}/E_{ν})^{T_{ν}}. The point P_{ν} is non-zero if and only if L′(ν, 1) ≠ 0.

Remark 3. Assumptions (8) and (9) do not reflect a serious limitation of our method of proof, but rather the fact that the main theorem of [BDP] on which it relies is only proved for imaginary quadratic fields of odd discriminant and with some restrictions on the conductor of ν. These assumptions could certainly be relaxed with some more work.

Remark 4. Equivalently, Theorem 2 could be stated by requiring that P_{ν} and ω_{ν} belong to (B_{ν}(K)⊗E_{ν})^{T_{ν}} and Ω^{1}(B_{ν}/K) respectively.

Remark 5. The methods used in the proof of Theorem 2 also give information about the special values ℒ_{p}(ν^{∗}) for Hecke characters ν of type (1 + j, −j) satisfying (6) with j ≥ 0. A discussion of this point will be taken up in future work.

Remark 6. The fact that our proof of Theorem 2 avoids the use of elliptic units raises the prospect of extending it to Hecke characters of more general CM fields.

Chow-Heegner points.

We now turn to the main goal of of this article: the study of generalisations of the Heegner point construction in which the role of Heegner divisors is played by null-homologous algebraic cycles of higher dimension.

We begin with a brief sketch of the classical picture which we aim to generalize. It is known thanks to [Wi], [TW], and [BCDT] that all elliptic curves over the rationals are modular. For an elliptic curve A of conductor N, this means that

(12)  L(A, s) = L(f, s),

where f(z) = ∑ a_{n}e^{2πinz} is a cusp form of weight 2 on the Hecke congruence group Γ_{0}(N). The modularity of A is established by showing that the p-adic Galois representation

V_{p}(A) := (lim_{←} A[p^{n}]) ⊗ ℚ_{p} = H_{et}^{1}(Ā, ℚ_{p})(1)

is a constituent of the first p-adic étale cohomology of the modular curve X_{0}(N). On the other hand, the Eichler-Shimura construction attaches to f an elliptic curve quotient A_{f} of the Jacobian J_{0}(N) of X_{0}(N) satisfying L(A_{f}, s) = L(f, s). In particular, the semisimple Galois representations V_{p}(A_{f}) and V_{p}(A) are isomorphic. It follows from Faltings’ proof of the Tate conjecture for abelian varieties over number fields that A is isogenous to A_{f}, and therefore there is a non-constant morphism

(13)  Φ : J_{0}(N)⟶A

of algebraic varieties over ℚ, inducing, for each F ⊃ ℚ a map Φ_{F} : J_{0}(N)(F)⟶A(F) on F -rational points.

A key application of Φ arises from the fact that X_{0}(N) is equipped with a distinguished supply of algebraic points corresponding to the moduli of elliptic curves with complex multiplication by an order in a quadratic imaginary field K. The images under Φℚ̄ of the degree 0 divisors supported on these points produce elements of A(ℚ̄) defined over abelian extensions of K, which include the so-called Heegner points. The Gross-Zagier formula [GZ] relates the canonical heights of these points to the central critical derivatives of L(A/K, s) and of its twists by abelian characters of K. This connection between algebraic points and Hasse-Weil L-series has led to the strongest known results on the Birch and Swinnerton-Dyer conjecture, most notably the theorem that

rank(A(ℚ)) = ord_{s=1} L(A, s) and #� (A/ℚ) < ∞, when ord_{s=1}(L(A, s)) ≤ 1,

which follows by combining the Gross-Zagier formula with a method of Kolyvagin (cf. [Gr2]). The theory of Heegner points is also the key ingredient in the proof of Theorems 1 and 2.

Given a variety X (defined over ℚ, say), let CH^{j}(X)(F) denote the Chow group of codimension j algebraic cycles on X defined over a field F modulo rational equivalence, and let CH^{j}(X)_{0}(F) denote the subgroup of null-homologous cycles. Write CH^{j}(X) and CH^{j}(X)_{0} for the corresponding functors on ℚ-algebras. Via the natural equivalence CH^{1}(X_{0}(N))_{0} = J_{0}(N), the map Φ of (13) can be recast as a natural transformation

(14)  Φ : CH^{1}(X_{0}(N))_{0}⟶A.

It is tempting to generalise (14) by replacing X_{0}(N) by a variety X over ℚ of dimension d > 1, and CH^{1}(X_{0}(N))_{0} by CH^{j}(X)_{0} for some 0 ≤ j ≤ d. Any element Π of the Chow group CH^{d+1−j}(X × A)(ℚ) induces a natural transformation

(15)  Φ : CH^{j}(X)_{0}⟶A

sending Δ ∈ CH^{j}(X)_{0}(F) to

(16)  Φ_{F} (Δ) := π_{A,∗}(π_{X}^{∗}(Δ̃) · Π̃),

where π_{X} and π_{A} denote the natural projections from X × A to X and A respectively. We are mainly interested in the case where X is a Shimura variety or is closely related to a Shimura variety. (For instance, when X is the universal object or a self-fold fiber product of the universal object over a Shimura variety of PEL type.) The variety X is then referred to as a modular variety and the natural transformation Φ is called the modular parametrisation of A attached to the pair (X, Π).

Modular parametrisations acquire special interest when CH^{j}(X)_{0}(ℚ̄) is equipped with a systematic supply of special elements, such as those arising from Shimura subvarieties of X. The images in A(ℚ̄) of such special elements under Φℚ̄ can be viewed as “higher-dimensional” analogues of Heegner points: they will be referred to as Chow-Heegner points. Given an elliptic curve A, it would be of interest to construct modular parametrisations to A in the greatest possible generality, study their basic properties, and explore the relations (if any) between the resulting systems of Chow-Heegner points and leading terms of L-series attached to A.

We develop this loosely formulated program in the simple but non-trivial setting where A is an elliptic curve with complex multiplication by an imaginary quadratic field K of odd discriminant −D, and X is a suitable family of 2r-dimensional abelian varieties fibered over a modular curve.

For the Introduction, suppose for simplicity that K has class number one and that A is the canonical elliptic curve over ℚ of conductor D^{2} attached to the Hecke character defined by

ν_{A}((a)) = ε_{K}(a mod \sqrt{−D})a.

(These assumptions will be significantly relaxed in the body of the paper.) Given a regular differential ω_{A} ∈ Ω^{1}(A/ℚ), let [ω_{A}] denote the corresponding class in the de Rham cohomology of A.

Fix an integer r ≥ 0, and consider the Hecke character ψ = ν_{A}^{r+1}. The binary theta series

θ_{ψ} := ∑_{𝔞⊂O_{K}} ν_{A}^{r+1}(𝔞)q^{𝔞𝔞̄}

attached to ψ is a modular form of weight (r+2) on a certain modular curve C (which is a quotient of X_{1}(D) or X_{0}(D^{2}) depending on whether r is odd or even), and has rational Fourier coefficients. Such a modular form gives rise to a regular differential (r +1)-form ω_{θ_{ψ}} on the rth Kuga-Sato variety over C, denoted W_{r}. Let [ω_{θ_{ψ}}] denote the class of ω_{θ_{ψ}} in the de Rham cohomology H_{dR}^{r+1}(W_{r}/ℚ). The classes of ω_{θ_{ψ}} and of the antiholomorphic (r+1)-form ω̄_{θ_{ψ}} generate the θ_{ψ}-isotypic component of H_{dR}^{r+1}(W_{r}/ℂ) under the action of the Hecke correspondences.

For all 1 ≤ j ≤ r + 1, let p_{j} : A^{r+1}⟶A denote the projection onto the j-th factor, and let

[ω_{A}^{r+1}] := p_{1}^{∗}[ω_{A}] ∧ ⋯ ∧ p_{r+1}^{∗}[ω_{A}] ∈ H_{dR}^{r+1}(A^{r+1}).

Our construction of Chow-Heegner points is based on the following conjecture which is formulated (for more general K, without the class number one hypothesis) in Section 3.

Conjecture 7. There is an algebraic cycle class Π^{?} ∈ CH^{r+1}(W_{r} × A^{r+1})(K) ⊗ ℚ satisfying

Π_{dR}^{?∗}([ω_{A}^{r+1}]) = [ω_{θ_{ψ}}],

where

Π_{dR}^{?∗} : H_{dR}^{r+1}(A^{r+1}/K)⟶H_{dR}^{r+1}(W_{r}/K)

is the map on de Rham cohomology induced by Π^{?}.

The rationale for Conjecture 7 is explained in Section 3.4, where it is shown to follow from the Tate or Hodge conjectures on algebraic cycles.

We next make the simple (but key) remark that the putative cycle Π^{?} is also an element of CH^{r+1}(X_{r} × A), where X_{r} is the (2r+1)-dimensional variety

X_{r} := W_{r} × A^{r}.

Viewed in this way, the cycle Π^{?} gives rise to a modular parametrisation

Φ^{?} : CH^{r+1}(X_{r})_{0}⟶A

as in (15). It is defined over K, and

(17)  Φ_{dR}^{?∗}(ω_{A}) = ω_{θ_{ψ}} ∧ η_{A}^{r},

where η_{A} is the unique element of H_{dR}^{1}(A/K) satisfying

(18)  [λ]^{∗}η_{A} = λ̄η, for all λ ∈ O_{K},  ⟨ω_{A}, η_{A}⟩ = 1.

The article [BDP] introduced and studied a collection of null-homologous, r-dimensional algebraic cycles on X_{r}, referred to as generalised Heegner cycles. These cycles, whose precise definition is recalled in Section 3.5, extend the notion of Heegner cycles on Kuga-Sato varieties considered in [Scho], [Ne] and [Zh]. They are indexed by isogenies φ : A⟶A′, and are defined over abelian extensions of K. It can be shown that they generate a subgroup of CH^{r+1}(X_{r})_{0}(K^{ab}) of infinite rank. The map Φ_{K^{ab}}^{?} should transform these generalised Heegner cycles into points of A(K^{ab}). It is natural to expect that the resulting collection {Φ_{K^{ab}}^{?}(Δ_{φ})}_{φ:A⟶A′} of Chow-Heegner points generates an infinite rank subgroup of A(K^{ab}), and that it gives rise to an “Euler system” in the sense of Kolyvagin. In the classical situation where r = 0, the variety X_{r} is just a modular curve and the existence of Π^{?} follows from Faltings’ proof of the Tate conjecture for curves. When r ≥ 1, the very existence of the collection of Chow-Heegner points relies, ultimately, on producing the algebraic cycle Π^{?} unconditionally.

Section 4 offers some theoretical evidence for the existence of Φ^{?} arising from p-adic methods. It rests on the fact that the following p-adic analogues of Φ_{F}^{?} can be constructed without invoking the Hodge or Tate conjectures.

(a) The map Φ_{F}^{et}: Let F be any field containing K. The Chow group CH^{r+1}(X_{r})_{0}(F) of null-homologous cycles is equipped with the p-adic étale Abel-Jacobi map over F:

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

where H^{1}(F, M) denotes the continuous Galois cohomology of a G_{F} := Gal(F̄/F)-module M. As is explained in Section 3.4, the Tate cycle over ℚ whose existence is used to justify Conjecture 7 gives rise to a G_{F} -equivariant projection

(20)  π : H^{2r+1}(\overline{X}_{r}, ℚ_{p})(r+1)⟶H_{et}^{1}(\overline{A}, ℚ_{p})(1) = V_{p}(A),

where V_{p}(A) is the p-adic Galois representation arising from the p-adic Tate module of A. As it stands, the maps π and Φ_{F}^{et} are only well-defined up to multiplication by ℚ_{p}^{×}. We normalise π by embedding F into a p-adic completion F_{p}, and requiring that the map π_{dR} : H_{dR}^{2r+1}(X_{r}/F_{p})⟶H_{dR}^{1}(A/F_{p}) obtained by applying to π the comparison functor between p-adic étale cohomology and deRham cohomology over p-adic fields satisfies

(21)  π_{dR}(ω_{θ_{ψ}} ∧ η_{A}^{r}) = ω_{A},

where ω_{θ_{ψ}} and ω_{A} are as in Conjecture 7, and η_{A} is defined in (18).

The maps (19) and (20) can be combined to give a map

(22)  Φ_{F}^{et} : CH^{r+1}(X_{r})(F)_{0}⟶H^{1}(F, V_{p}(A)),

which is the counterpart of the conjectural map Φ_{F}^{?} in p-adic étale cohomology. More precisely, with π chosen to satisfy (21), the map Φ_{F}^{et} is related to Φ_{F}^{?} (when the latter can be shown to exist) by the commutative diagram

(23)  \xymatrix{ & A(F) ⊗ ℚ \ar[d]^{δ} \\ CH^{r+1}(X_{r})_{0}(F) \ar@{-->}[ur]^{Φ_{F}^{(?)}} \ar[r]^{Φ_{F}^{et}} & H^{1}(F, V_{p}(A)),}

where

(24)  δ : A(F) ⊗ ℚ⟶H^{1}(F, V_{p}(A))

is the projective limit of the connecting homomorphisms arising in the p^{n}-descent exact sequences of Kummer theory.

(b) The map Φ_{F}^{(p)}: When F is a number field, (23) suggests that the image of Φ_{F}^{et} is contained in the Selmer group of A over F, and this can indeed be shown to be the case. This means that the image of Φ_{F_{v}}^{et} is contained in the images of the local connecting homomorphisms δ_{v} : A(F_{v})⟶H^{1}(F_{v}, V_{p}(A)) for all the completions of F. In particular, replacing F by its p-adic completion F_{p}, we can define the map Φ_{F}^{(p)} by the commutativity of the following local counterpart of the diagram (23):

(25)  \xymatrix{ & A(F_{p}) ⊗ ℚ \ar[d]^{δ} \\ CH^{r+1}(X_{r})_{0}(F_{p}) \ar[ur]^{Φ_{F}^{(p)}} \ar[r]^{Φ_{Fp}^{et}} & H^{1}(F_{p}, V_{p}(A)).}

As will be explained in more detail in Section 4, the map Φ_{F}^{(p)} can also be defined by p-adic integration, via the comparison theorems between the p-adic étale cohomology and the de Rham cohomology of varieties over p-adic fields.

The main result of Chapter 4 relates the Selmer classes of the form Φ_{F}^{et}(Δ) when F is a global field and Δ is a generalised Heegner cycle, to global points in A(F). We will only state a special case of the main result, postponing the more general statement to Section 4.2. Assume for Theorem 8 below that the field K has odd discriminant, that the sign in the functional equation for L(ν_{A}, s) is −1, so that the Hasse-Weil L-series L(A/ℚ, s) = L(ν_{A}, s) vanishes to odd order at s = 1, and that the integer r is odd. In that case, the theta series θ_{ψ} belongs to the space S_{r+2}(Γ_{0}(D), ε_{K}) of cusp forms on Γ_{0}(D) of weight r + 2 and character ε_{K} := (\frac{·}{D} ). In particular, the variety W_{r} is essentially the rth Kuga-Sato variety over the modular curve X_{0}(D). Furthermore, the L-series L(ν_{A}^{2r+1}, s) has sign 1 in its functional equation, and L(ν_{A}^{2r+1}, s) therefore vanishes to even order at the central point s = r + 1.

Theorem 8. Let Δ be the generalised Heegner cycle in CH^{r+1}(X_{r})_{0}(K) attached to the identity isogeny 1 : A⟶A. The cohomology class Φ_{K}^{et}(Δ) belongs to δ(A(K) ⊗ ℚ). More precisely, there is a point P_{D} ∈ A(K) ⊗ ℚ (depending on D but not on r) such that

Φ_{K}^{et}(Δ) = \sqrt{−D} · m_{D,r} · δ(P_{D}),

where m_{D,r} ∈ ℤ is given by

m_{D,r}^{2} = \frac{2r!(2π\sqrt{D})^{r}}{Ω(A)^{2r+1}} L(ν_{A}^{2r+1}, r + 1),

and Ω(A) is a complex period attached to A. The point P_{D} is of infinite order if and only if

L′(ν_{A}, 1) ≠ 0.

Remark 9. When L(A, s) has a simple zero at s = 1, it is known a priori that the Selmer group Sel_{p}(A/K) is of rank one over K ⊗ ℚ_{p}, and agrees with δ(A(K) ⊗ ℚ_{p}). It follows directly that

Φ_{K}^{et}(Δ) belongs to δ(A(K) ⊗ ℚ_{p}).

The first part of Theorem 8 is stronger in that it involves the rational vector space A(K)⊗ℚ rather than its p-adification. This stronger statement is not a formal consequence of the one-dimensionality of the Selmer group. Indeed, its proof relies on invoking Theorem 2 after relating the local point Φ_{K_{p}}^{(p)}(Δ) ∈ A(K_{p}) to the special value ℒ_{p}(ν_{A}^{∗}) that arises in that theorem.

Section 5.1 describes a complex homomorphism

Φ_{ℂ} : CH^{r+1}(X_{r})_{0}(ℂ)⟶A(ℂ)

which is defined analytically by integration of differential forms on X_{r}(ℂ), without invoking Conjecture 7, but agrees with Φ_{ℂ}^{?} when the latter exists. This map is defined from the complex Abel-Jacobi map on cycles introduced and studied by Griffiths and Weil, and is the complex analogue of homomorphism Φ_{F}^{(p)} defined in (25). The existence of the global map Φ_{K}^{?} predicted by the Hodge or Tate conjecture would imply the following algebraicity statement:

Conjecture 10. Let H be a subfield of K^{ab} and let Δ_{φ} ∈ CH^{r+1}(X_{r})_{0}(H) be a generalised Heegner cycle defined over H. Then (after fixing an an embedding of H into ℂ),

Φ_{ℂ}(Δ_{φ}) belongs to A(H),

and

Φ_{ℂ}(Δ_{φ}^{σ}) = Φ_{ℂ}(Δ_{φ})^{σ} for all σ ∈ Gal(H/K).

While ostensibly weaker than Conjecture 7, Conjecture 10 has the virtue of being more readily amenable to experimental verification. Section 5 explains how the images of generalised Heegner cycles under Φ_{ℂ} can be computed numerically to high accuracy, and illustrates, for a few such Δ_{φ}, how the points Φ_{ℂ}(Δ_{φ}) can be recognized as algebraic points defined over the predicted class fields. In particular, extensive numerical verifications of Conjecture 10 are carried out, for fairly large values of r. This conjecture appears to lie deeper than its p-adic counterpart, and we were unable to provide any theoretical evidence for it beyond the fact that it follows from the Hodge or Tate conjectures. It might be argued that calculations of the sort that are performed in Section 5 provide independent numerical confirmation of the these conjectures for certain specific Hodge and Tate cycles on the (2r + 2)-dimensional varieties W_{r} × A^{r+1}, for which the corresponding algebraic cycles seem hard to produce unconditionally.

Conventions regarding number fields and embeddings: Throughout this article, all number fields that arise are viewed as embedded in a fixed algebraic closure ℚ̄ of ℚ. A complex embedding ℚ̄⟶ℂ and p-adic embeddings ℚ̄⟶ℂ_{p} for each rational prime p are also fixed from the outset, so that any finite extension of ℚ is simultaneously realised as a subfield of ℂ and of ℂ_{p}.

1. Hecke characters and periods

1.1. Algebraic Hecke characters. Let K and E be number fields. Given a ℤ-linear combination

ϕ = ∑_{σ} n_{σ}σ ∈ ℤ[Hom(K, ℚ̄]

of embeddings of K into ℚ̄, we define

α^{ϕ} := ∏_{σ}(σα)^{n_{σ}},

for all α ∈ K^{×}. Let I_{𝔣} denote the group of fractional ideals of K which are prime to a given integral ideal 𝔣 of K, and let

J_{𝔣} := {(α) such that α >> 0 and α − 1 ∈ 𝔣} ⊂ I_{𝔣}.

Definition 1.1. An E-valued algebraic Hecke character (or simply Hecke character) of K of infinity type ϕ on I_{𝔣} is a homomorphism

χ : I_{𝔣} → E^{×}

such that

(26)  χ((α)) = α^{ϕ}, for all (α) ∈ J_{𝔣}.

The largest integral ideal 𝔣 satisfying (26) is called the conductor of χ, and is denoted 𝔣_{χ}.

The most basic examples of algebraic Hecke characters are the norm characters on ℚ and on K respectively, which are given by

N(a) = |a|, N_{K} := N ∘ N_{ℚ}^{K}.

Note that the infinity type ϕ of a Hecke character χ must be trivial on all totally positive units congruent to 1 mod 𝔣. Hence the existence of such a χ implies there is an integer w(χ) (called the weight of χ or of ϕ) such that for any choice of embedding of \overline{ℚ} into ℂ,

n_{σ} + nσ̄ = w(χ), for all σ ∈ hom(K, ℚ̄).

Let U′_{𝔣} ⊂ 𝔸_{K}^{×} be the subgroup defined by

U′_{𝔣} := \{(x_{v}) ∈ 𝔸_{K}^{×} such that \begin{matrix}x_{v} ≡ 1 (mod 𝔣), for all v|𝔣, \\ x_{v} > 0, for all real v \end{matrix} \} ,

and let U_{𝔣} denote its maximal compact subgroup:

U_{𝔣} := {(x_{v}) ∈ U′_{𝔣} such that x_{v} ∈ O_{K_{v}}^{×}, for all non-archimedean v}.

A Hecke character χ may also be viewed as a character on 𝔸_{K}^{×}/U_{𝔣} (denoted by the same symbol by a common abuse of notation),

(27)  χ : 𝔸_{K}^{×}/U_{𝔣} → E^{×},  satisfying  χ|_{K^{×}} = ϕ.

To wit, given x ∈ 𝔸_{K}^{×}, we define χ(x) by choosing α ∈ K^{×} such that αx belongs to U′_{𝔣}, and setting

(28)  χ(x) = χ(i(αx))ϕ(α)^{−1},

where the symbol i(x) denotes the fractional ideal of K associated to x. This definition is independent of the choice of α by (26). In the opposite direction, given an idèle character χ of conductor 𝔣 as in (27), we can set

χ(𝔞) = χ(x), for any x ∈ U′_{𝔣} such that i(x) = 𝔞.

The subfield of E generated by the values of χ on I_{𝔣} is easily seen to be independent of the choice of 𝔣 and will be denoted E_{χ}.

The central character of a Hecke character of K is defined as follows:

Definition 1.2. The central character of a Hecke character η of K is the finite order character of ℚ given by

η|_{𝔸_{ℚ}^{×}} = ε_{η} · N^{w(η)}.

The infinity type ϕ defines a homomorphism Res_{K,ℚ}(G_{m}) → Res_{E/ℚ} G_{m} of algebraic groups and therefore induces a homomorphism

ϕ_{𝔸} : 𝔸_{K}^{×} → 𝔸_{E}^{×}

on adelic points. Given a Hecke character χ with values in E and a place λ of E (either finite or infinite), we may use ϕ_{𝔸} to define an idèle class character

χ_{λ} : 𝔸_{K}^{×}/K^{×} → E_{λ}^{×},

by setting

χ_{λ}(x) = χ(x)/ϕ_{𝔸}(x)_{λ}.

If λ is an infinite place, the character χ_{λ} is a Grossencharacter of K of type A_{0}. If λ is a finite place, then χ_{λ} factors through G_{K}^{ab} and gives a Galois character valued in E_{λ}^{×}, satisfying

χ_{λ}(Frob_{𝔭}) = χ(𝔭)

for any prime ideal 𝔭 of K not dividing 𝔣λ.

Definition 1.3. Let E = ∏_{i} E_{i} be a product of number fields. An E-valued algebraic Hecke character is a character

χ : I_{𝔣} → E^{×}

whose projection to each component E_{i} is an algebraic Hecke character in the sense defined above.

The L-functions attached to any algebraic Hecke character χ are defined by

L(χ, s) = ∏_{𝔭} (1 − \frac{χ(𝔭)}{N𝔭^{s}})^{−1} , L_{𝔣}(χ, s) = ∏_{𝔭∤𝔣} (1 − \frac{χ(𝔭)}{N𝔭^{s}})^{−1} .

Note that L(χ, s) = L_{𝔣_{χ}}(χ, s) when χ is a primitive Hecke character.

Remark 1.4. The definition of L(χ, s) as a ℂ-valued function relies on the fact that E_{χ} is given as a subfield of ℂ.

1.2. Abelian varieties associated to characters of type (1, 0). In this section, we limit the discussion to the case where K is an imaginary quadratic field. Let τ : K ↦ ℂ be the given complex embedding of K. A Hecke character of infinity type ϕ = n_{τ} τ + nτ̄τ̄ will also be said to be of infinity type (n_{τ} , nτ̄).

Let ν be a Hecke character of K of infinity type (1, 0) and conductor 𝔣_{ν}, let E_{ν} ⊃ K denote the subfield of ℚ̄ generated by its values, and let T_{ν} be the ring of integers of E_{ν}. The Hecke character ν gives rise to a compatible system of one-dimensional ℓ-adic representations of G_{K} with values in (E ⊗ ℚ_{ℓ})^{×}, denoted ρ_{ν,ℓ}, satisfying

ρ_{ν,ℓ}(σ_{𝔞}) = ν(𝔞), for all 𝔞 ∈ I_{𝔣_{ν}ℓ},

where σ_{𝔞} ∈ Gal(K̄/K) denotes frobenius conjugacy class attached to 𝔞. The theory of complex multiplication realises the representations ρ_{ν,ℓ} on the division points of so-called CM abelian varieties, which are defined as follows.

Definition 1.5. A CM abelian variety over K is a pair (B, E) where

(1) B is an abelian variety over K;

(2) E is a product of CM fields equipped the structure of a K-algebra and an inclusion

i : E⟶ End_{K}(B) ⊗ ℚ,

satisfying dim_{K}(E) = dim B;

(3) for all λ ∈ K ⊂ E, the endomorphism i(λ) acts on the cotangent space Ω^{1}(B/K) as multiplication by λ.

The abelian varieties (B, E) over K with complex multiplication by a fixed E form a category denoted CM_{K,E} in which a morphism from B_{1} to B_{2} is a morphism j : B_{1}⟶B_{2} of abelian varieties over K for which the diagrams

\xymatrix{B_{1} \ar[r]^{j} \ar[d]^{e} & B_{2} \ar[d]^{e} \\ B_{1} \ar[r]^{j} & B_{2}}

commute, for all e ∈ E which belong to both End_{K}(B_{1}) and End_{K}(B_{2}). An isogeny in CM_{K,E} is simply a morphism in this category arising from an isogeny on the underlying abelian varieties.

If (B, E) is a CM abelian variety, its endomorphism ring over K contains a finite index subring T^{0} of the integral closure T of ℤ in E. After replacing B by the K-isogenous abelian variety hom_{T_{0}}(T, B), we can assume that End_{K}(B) contains T. This assumption, which is occasionally convenient, will consistently be made from now on.

Let (B, E) be a CM-abelian variety with E a field, and let E′ ⊃ E be a finite extension of E with ring of integers T′. The abelian variety B ⊗_{T} T′ is defined to be the variety whose L-rational points, for any L ⊃ K, are given by

(B ⊗_{T} T′)(L) = (B(ℚ̄) ⊗_{T} T′)^{Gal(ℚ̄/L)}.

This abelian variety is equipped with an action of T′ by K-rational endomorphisms, described by multiplication on the right, and therefore (B ⊗_{T} T′, E′) is an object of CM_{K,E′}. Note that B ⊗_{T} T′ is isogenous to t := dim_{E}(E′) copies of B, and that the action of T on B ⊗_{T} T′ agrees with the “diagonal” action of T on B^{t}.

Let ℓ be a rational prime. For each CM abelian variety (B, E), let

T_{ℓ}(B) := lim_{←,n} B[ℓ^{n}](K̄), V_{ℓ}(B) := T_{ℓ}(B) ⊗_{ℤ_{ℓ}} ℚ_{ℓ}

be the ℓ-adic Tate module and ℓ-adic representation of G_{K} attached to B. The ℚ_{ℓ}-vector space V_{ℓ}(B) is a free E ⊗ ℚ_{ℓ}-module of rank one via the action of E by endomorphisms. The natural action of G_{K} := Gal(K̄/K) on V_{ℓ}(B) commutes with this E ⊗ ℚ_{ℓ}-action, and the collection {V_{ℓ}(B)} thus gives rise to a compatible system of one-dimensional ℓ-adic representations of G_{K} with values in (E ⊗ℚ_{ℓ})^{×}, denoted ρ_{B,ℓ}. We note in passing that for any extension E′ ⊃ E where T′ the integral closure of T in E′,

T_{ℓ}(B ⊗_{T} T′) = T_{ℓ}(B) ⊗_{T} T′, V_{ℓ}(B ⊗_{T} T′) = V_{ℓ}(B) ⊗_{E} E′.

The following result is due to Casselman (cf. Theorem 6 of [Shi]).

Theorem 1.6. Let ν be a Hecke character of K of type (1, 0) as above, and let ρ_{ν,ℓ} be the associated one-dimensional ℓ-adic representation with values in (E_{ν} ⊗ ℚ_{ℓ}). Then

(1) There exists a CM abelian variety (B_{ν}, E_{ν}) satisfying

ρ_{B_{ν},ℓ} ≃ ρ_{ν,ℓ}.

(2) The CM abelian variety B_{ν} is unique up to isogeny over K. More generally, if (B, E) is any CM abelian variety with E ⊃ E_{ν} satisfying ρ_{B,ℓ} ≃ ρ_{ν,ℓ} ⊗_{E_{ν}} E as (E ⊗ ℚ_{ℓ})[G_{K}]-modules, then there is an isogeny in CM_{K,E} from B to B_{ν} ⊗_{T_{ν}} T.

Let ψ be a Hecke character of infinity type (1, 0), and let χ be a finite order Hecke character. In comparing the abelian varieties B_{ψ} and B_{ψχ^{−1}} attached to these two characters, it is useful to introduce a CM abelian variety B_{ψ,χ} over K, which we now proceed to describe.

Let E_{ψ,χ} be the subfield of ℚ̄ generated by E_{ψ} and E_{χ}, and let T_{ψ,χ} ⊂ E_{ψ,χ} be its ring of integers. We also write H_{χ} for the abelian extension of K which is cut out by χ viewed as a Galois character of G_{K}. Consider first the abelian variety over K with endomorphism by T_{ψ,χ}:

B_{ψ,χ}^{0} := B_{ψ} ⊗_{T_{ψ}} T_{ψ,χ}.

The natural inclusion i_{ψ} : T_{ψ}⟶T_{ψ,χ} induces a morphism i : B_{ψ}⟶B_{ψ,χ}^{0} with finite kernel, which is compatible with the T_{ψ}-actions on both sides and is given by

i(P) = P ⊗ 1.

Lemma 1.7. Let F be any number field containing E_{ψ,χ}. The restriction map i^{∗} induces an isomorphism

(29)  i^{∗} : Ω^{1}(B_{ψ,χ}^{0}/F)^{T_{ψ,χ}}⟶Ω^{1}(B_{ψ}/F)^{T_{ψ}}

of one-dimensional F -vector spaces.

Proof. The fact that B_{ψ,χ}^{0} and B_{ψ} are CM abelian varieties over F implies that source and target in (29) are both one-dimensional over F. To see that i^{∗} is injective, let ω be an element of ker(i^{∗}) ∩ Ω^{1}(B_{ψ,χ}^{0}/F)^{T_{ψ,χ}}. Then since ω is stable under the action of T_{ψ,χ} by endomorphisms, it follows that

ω belongs to ∩_{λ∈T_{ψ,χ}} [λ] ker(i^{∗}) = 0.

Hence ω = 0 and the lemma follows. □

We can now denote by ω_{ψ,χ}^{0} ∈ Ω^{1}(B_{ψ,χ}^{0}/ℚ̄)^{T_{ψ,χ}} the unique regular differential satisfying

(30)  i^{∗}(ω_{ψ,χ}) = ω_{ψ}, where ω_{ψ} ∈ Ω^{1}(B_{ψ}/E_{ψ})^{T_{ψ}}.

It follows from Lemma 1.7 that ω_{ψ,χ}^{0} exists and is unique (once ω_{ψ} has been chosen), and that ω_{ψ,χ}^{0} belongs to Ω^{1}(B_{ψ,χ}^{0}/E_{ψ,χ}).

The character χ^{−1} : Gal(H_{χ}/K)⟶T_{χ}^{×} can be viewed as a one-cocycle in

H^{1}(Gal(H_{χ}/K), T_{ψ,χ}^{×}) ⊂ H^{1}(Gal(H_{χ}/K), Aut(B_{ψ,χ}^{0})).

Let

(31)  B_{ψ,χ} := (B_{ψ,χ}^{0})^{χ^{−1}}

denote the twist of B_{ψ,χ}^{0} by this cocycle. There is a natural identification B_{ψ,χ}^{0}(K̄) = B_{ψ,χ}(K̄) of sets, arising from an isomorphism of varieties over H_{χ}, where H_{χ} is the extension of K cut out by χ. The actions of G_{K} on B_{ψ,χ}^{0}(K̄) and B_{ψ,χ}(K̄), denoted ∗_{0} and ∗ respectively, are related by

(32)  σ ∗ P = (σ ∗_{0} P) ⊗ χ^{−1}(σ),  for all σ ∈ G_{K}.

In particular, for any L ⊃ K, we have:

(33)  B_{ψ,χ}(L) = {P ∈ B_{ψ}(ℚ̄) ⊗_{T_{ψ}} T_{ψ,χ} such that σP = P ⊗ χ(σ), ∀σ ∈ Gal(ℚ̄/L)} .

Likewise, the natural actions of G_{K} on Ω^{1}(B_{ψ,χ}^{0}/K̄) and on Ω^{1}(B_{ψ,χ}/K̄) are related by

(34)  σ ∗ ω = [χ^{−1}(σ)]^{∗}(σ ∗_{0} ω)  for all σ ∈ G_{K}.

The isomorphism of B_{ψ,χ}^{0} and B_{ψ,χ} as CM abelian varieties over H_{χ} gives natural identifications

Ω^{1}(B_{ψ,χ}^{0}/H_{χ}) = Ω^{1}(B_{ψ,χ}/H_{χ}), Ω^{1}(B_{ψ,χ}^{0}/E′_{ψ,χ})^{T_{ψ,χ}} = Ω^{1}(B_{ψ,χ}/E′_{ψ,χ})^{T_{ψ,χ}},

where E′_{ψ,χ} denotes the subfield of ℚ̄ generated by H_{χ} and E_{ψ,χ}. Let ω_{ψ,χ}^{0} and ω_{ψ,χ} be E_{ψ,χ} vector space generators of Ω^{1}(B_{ψ,χ}^{0}/E_{ψ,χ})^{T_{ψ,χ}} and Ω^{1}(B_{ψ,χ}/E_{ψ,χ})^{T_{ψ,χ}} respectively. Since they both generate Ω^{1}(B_{ψ,χ}/E′_{ψ,χ})^{T_{ψ,χ}} as an E′_{ψ,χ}-vector space, they necessarily differ by a non-zero scalar in E′_{ψ,χ}.

To spell out the relation between ω_{ψ,χ}^{0} and ω_{ψ,χ} more precisely, it will be useful to introduce the notion of a generalised Gauss sum attached to any finite order character χ of G_{K}. Given such a character, let E ⊂ ℚ̄ denote the field generated by the values of χ, and let

E{χ} := {λ ∈ EH_{χ} such that λ^{σ} = χ(σ)λ, ∀σ ∈ Gal(EH_{χ}/E)}.

This set is a one-dimensional E-vector space in a natural way. It is not closed under multiplication, but E{χ_{1}} · E{χ_{2}} = E{χ_{1}χ_{2}}.

Definition 1.8. An E-vector space generator of E{χ} is called a Gauss sum attached to the character χ, and is denoted 𝔤(χ).

By definition, the Gauss sum 𝔤(χ) belongs to E{χ} ∩ (EH_{χ})^{×}, but is only well-defined up to multiplication by E^{×}.

The following lemma pins down the relationship between the differentials ω_{ψ,χ}^{0} and ω_{ψ,χ}.

Lemma 1.9. For all Hecke characters ψ and χ as above,

ω_{ψ,χ} = 𝔤(χ)ω_{ψ,χ}^{0} (mod E_{ψ,χ}^{×}).

Proof. Let λ ∈ (H_{χ}E_{ψ,χ})^{×} be the scalar satisfying

(35)  ω_{ψ,χ} = λω_{ψ,χ}^{0}.

Since ω_{ψ,χ} is an E_{ψ,χ}-rational differential on B_{ψ,χ}, for all σ ∈ Gal(K̄/E_{ψ,χ}) we have

(36)  ω_{ψ,χ} = σ ∗ ω_{ψ,χ} = [χ^{−1}(σ)]^{∗}σ ∗_{0} ω_{ψ,χ} = χ^{−1}(σ)λ^{σ}ω_{ψ,χ}^{0},

where the second equality follows from equation (34) and the last from the fact that the differential ω_{ψ,χ}^{0} belongs to Ω^{1}(B_{ψ,χ}^{0}/E_{ψ,χ})^{T_{ψ,χ}}. Comparing (35) and (36) gives λ^{σ} = χ(σ)λ, and hence λ = 𝔤(χ) (mod E_{ψ,χ}^{×}). The lemma follows. □

The following lemma relates the abelian varieties B_{ψ,χ} and B_{ν}, where ν = ψχ^{−1}.

Lemma 1.10. There is an isogeny defined over K:

i_{ν} : B_{ψ,χ}⟶B_{ν} ⊗_{T_{ν}} T_{ψ,χ}

which is compatible with the action of T_{ψ,χ} by endomorphisms on both sides.

Proof. The pair (B_{ψ,χ}^{0}, E_{ψ,χ}) is a CM abelian variety having ψ (viewed as taking values in E_{ψ,χ}) as its associated Hecke character. The Hecke character attached to the Galois twist B_{ψ,χ} is therefore ψχ^{−1} = ν. The second part of Theorem 1.6 implies that B_{ψ,χ} and B_{ν} ⊗_{T_{ν}} T_{ψ,χ} are isogenous over K as CM abelian varieties, and the lemma follows. □

1.3. Complex periods and special values of L-functions. This section attaches certain periods to the quadratic imaginary field K and to Hecke characters of this field.

We begin by defining a complex period attached to K. This period depends on the following choices:

(1) An elliptic curve A with complex multiplication by O_{K}, defined over a finite extension F of K. (Note that F necessarily contains the Hilbert class field of K.)

(2) A regular differential ω_{A} ∈ Ω^{1}(A/F).

(3) A non-zero element γ of H_{1}(A(ℂ), ℚ).

The complex period attached to this data is defined by

(37)  Ω(A) := \frac{1}{2πi} ∫_{γ} ω_{A}  (mod ℚ̄^{×}) (mod F^{×}).

Note that Ω(A) depends on the pair (ω, γ). A different choice of ω or γ has the effect of multiplying Ω(A) by a scalar in F^{×}, and therefore Ω(A) can be viewed as a well-defined element of ℂ^{×}/F^{×}.

For any Hecke character ψ of K, recall that ψ^{∗} is the Hecke character defined as in the Introduction by ψ^{∗}(x) = ψ(x̄). Suppose that ψ is of infinity type (1, 0), and let E_{ψ} ⊂ ℚ̄ ⊂ ℂ denote the field generated by the values of ψ (or, equivalently, ψ^{∗}). Choose (arbitrary) non-zero elements

ω_{ψ} ∈ Ω^{1}(B_{ψ}/E_{ψ})^{T_{ψ}}, γ ∈ H_{1}(B_{ψ}(ℂ), ℚ),

where B_{ψ} is the CM abelian variety attached to ψ by Theorem 1.6, and Ω^{1}(B_{ψ}/E_{ψ})^{T_{ψ}} is defined in equation (10) of the Introduction. The period Ω(ψ^{∗}) attached to ψ^{∗} is defined by setting

Ω(ψ^{∗}) = \frac{1}{2πi} ∫_{γ} ω_{ψ} (mod E_{ψ}^{×}).

Note that the complex number Ω(ψ^{∗}) does not depend, up to multiplication by E_{ψ}^{×}, on the choices of ω_{ψ} and γ that were made to define it.

Lemma 1.11. If ψ is a Hecke character of infinity type (1, 0), and χ is a finite order character, then

(38)  Ω(ψ^{∗}χ) = Ω(ψ^{∗})𝔤(χ^{∗})^{−1} (mod E_{ψ,χ}^{×}).

Proof. Choose a non-zero generator γ of H_{1}(B_{ψ,χ}^{0}(ℂ), ℚ) = H_{1}(B_{ψ,χ}(ℂ), ℚ) (viewed as a one-dimensional E_{ψ,χ} vector space via the endomorphism action). By definition,

Ω((ψχ^{−1})^{∗}) = ∫_{γ} ω_{ψ,χ} = 𝔤(χ) ∫_{γ} ω_{ψ,χ}^{0} = 𝔤(χ)Ω(ψ^{∗}) (mod E_{ψ,χ}^{×}),

where the second equality follows from Lemma 1.9. The result now follows after substituting χ^{∗−1} for χ. □

One can also attach a period Ω(ψ^{∗}) to an arbitrary Hecke character of K following the article [GS]. (See also Sections 3.1 and 3.2.) These more general periods are known to satisfy the following multiplicativity relations: see Section 1 of [Har] for instance.

Proposition 1.12. Let ψ be a Hecke character of infinity type (k, j). Then (1) The ratio

\frac{Ω(ψ^{∗})}{(2πi)^{j}Ω(A)^{k−j}}

is algebraic.

(2) For all ψ and ψ′,

Ω(ψψ′) = Ω(ψ)Ω(ψ′) (mod E_{ψ,ψ′}^{×}),

where E_{ψ,ψ′} is the subfield of ℚ̄ generated by E_{ψ} and E_{ψ′}.

The following theorem is due to Goldstein and Schappacher [GS] (and Blasius for general CM fields).

Theorem 1.13. Suppose that ψ has infinity type (k, j) with k > j, and that m is a critical integer for L(ψ^{−1}, s). Then

\frac{L(ψ^{−1}, m)}{(2πi)^{m}Ω(ψ^{∗})} belongs to E_{ψ},

and for all τ ∈ Gal(E_{ψ}/K),

(\frac{L(ψ^{−1}, m)}{(2πi)^{m}Ω(ψ^{∗})})^{τ} = \frac{L((ψ^{−1})^{τ} , m)}{(2πi)^{m}Ω((ψ^{∗})^{τ})}.

1.4. p-adic periods. Fix a prime p that splits in K. We will need p-adic analogs of the periods Ω(A) and Ω(ν^{∗}). One way to define these p-adic periods is to use the comparison isomorphism between p-adic étale cohomology over p-adic fields and de Rham cohomology, as hinted in Blasius’s article [Bl]. It is also possible to supply a more direct definition, as will be done in this section.

The p-adic analogue Ω_{p}(A) of Ω(A) is obtained by considering the base change A_{ℂ_{p}} of A to ℂ_{p} (via our fixed embedding of F into ℂ_{p}). Assume that A has good reduction at the maximal ideal of O_{ℂ_{p}}, i.e., that A_{ℂ_{p}} extends to a smooth proper model A_{O_{ℂp}} over O_{ℂ_{p}}. The p-adic completion Â_{O_{ℂp}} of A along its special fiber is isomorphic to Ĝ_{m}. Following [deS] II, §4.4, choose an isomorphism ι_{p} : Â⟶Ĝ_{m} over O_{ℂ_{p}}, and define Ω_{p}(A) ∈ ℂ_{p}^{×} by the rule

(39)  ω_{A} = Ω_{p}(A) · ι_{p}^{∗}(du/u),

where u is the standard coordinate on Ĝ_{m}. The invariant Ω_{p}(A) ∈ ℂ_{p}^{×} thus defined depends on the choices of ω_{A} and ι_{p}, but only up to multiplication by a scalar in F^{×}. Observe also that Ω(A) and Ω_{p}(A) each depend linearly in the same way on the choice of the global differential ω_{A}.

The p-adic period Ω_{p}(A) can be used to define p-adic analogues of the complex periods Ω(ν) that appear in the statement of Theorem 1.13.

Definition 1.14. The p-adic period attached to a Hecke character ν of type (1, 0), denoted Ω_{p}(ν^{∗}), is defined to be

Ω_{p}(ν^{∗}) := Ω_{p}(A) · \frac{Ω(ν^{∗})}{Ω(A)}.

More generally, for any character ν of infinity type (k, j), we define

Ω_{p}(ν^{∗}) := Ω_{p}(A)^{k−j} · \frac{Ω(ν^{∗})}{(2πi)^{j}Ω(A)^{k−j}} .

It can be seen from this definition that the period Ω_{p}(ν), like its complex counterpart Ω(ν), is welldefined up to multiplication by a scalar in E_{ν}^{×}. The following p-adic analogue of Lemma 1.11 is a direct consequence of this lemma combined with the definition of Ω_{p}(ψ):

Lemma 1.15. If ψ is a Hecke character of infinity type (1, 0), and χ is a finite order character, then

(40)  Ω_{p}(ψ^{∗}χ) = Ω_{p}(ψ^{∗})𝔤(χ^{∗})^{−1} (mod E_{ψ,χ}^{×}).

Likewise, Proposition 1.12 implies:

Proposition 1.16. Let ψ be a Hecke character of infinity type (k, j). Then (1) The ratio

\frac{Ω_{p}(ψ^{∗})}{(2πi)^{j}Ω_{p}(A)^{k−j}}

is algebraic.

(2) For all ψ and ψ′,

(41)  Ω_{p}(ψψ′) = Ω_{p}(ψ)Ω_{p}(ψ′) (mod E_{ψ,ψ′}^{×}).

2. p-adic L-functions and Rubin’s formula

2.1. The Katz p-adic L-function. Throughout this chapter, we will fix a prime p that is split in K. Let 𝔠 be an integral ideal of K which is prime to p, and let Σ(𝔠) denote the set of all Hecke characters of K of conductor dividing 𝔠. Denote by 𝔭 the prime above p corresponding to the chosen embedding K ↪ ℚ̄_{p}.

A character ν ∈ Σ(𝔠) is called a critical character if L(ν^{−1}, 0) is a critical value in the sense of Deligne, i.e., if the Γ-factors that arise in the functional equation for L(ν^{−1}, s) are non-vanishing and have no poles at s = 0. The set Σ_{crit}(𝔠) of critical characters can be expressed as the disjoint union

Σ_{crit}(𝔠) = Σ_{crit}^{(1)}(𝔠) ∪ Σ_{crit}^{(2)}(𝔠),

where

Σ_{crit}^{(1)}(𝔠) = {ν ∈ Σ(𝔠) of type (ℓ_{1}, ℓ_{2}) with ℓ_{1} ≤ 0, ℓ_{2} ≥ 1} ,
Σ_{crit}^{(2)}(𝔠) = {ν ∈ Σ(𝔠) of type (ℓ_{1}, ℓ_{2}) with ℓ_{1} ≥ 1, ℓ_{2} ≤ 0} .

The possible infinity types of Hecke characters in these two critical regions are sketched in Figure 1. Note in particular that when 𝔠 = 𝔠̄, the regions Σ_{crit}^{(1)}(𝔠) and Σ_{crit}^{(2)}(𝔠) are interchanged by the involution ν ↦ ν^{∗}, where ν^{∗} is defined as in the Introduction by ν^{∗}(a) = ν(ā), with a ↦ ā denoting the conjugation on 𝔸_{K}.

The set Σ_{crit}(𝔠) is endowed with a natural p-adic topology arising from the compact open topology on the space of functions on a certain subset of 𝔸_{K}^{×}, as described in Section 5.4 of [BDP]. The subsets Σ_{crit}^{(1)}(𝔠) and Σ_{crit}^{(2)}(𝔠) are dense in the completion Σ̂_{crit}(𝔠) relative to this topology.

Recall that 𝔭 is the prime above p induced by our chosen embedding of K into ℂ_{p}. The following proposition on the existence of the p-adic L-function is due to Katz. The statement below is a restatement of [deS] (II, Thm. 4.14) with a minor correction.

Proposition 2.1. There exists a p-adic analytic function ν ↦ ℒ_{p}(ν) (valued in ℂ_{p}) on Σ̂_{crit}(𝔠) which is determined by the interpolation property:

(42)  \frac{ℒ_{p}(ν)}{Ω_{p}(A)^{ℓ_{1}−ℓ_{2}}} = (\frac{\sqrt{D}}{2π} )^{ℓ_{2}} (ℓ_{1} − 1)!(1 − ν(𝔭)/p)(1 − ν^{−1}(𝔭̄))\frac{L_{𝔠}(ν^{−1}, 0)}{Ω(A)^{ℓ_{1}−ℓ_{2}}},

for all critical characters ν ∈ Σ_{crit}^{(2)}(𝔠) of infinity type (ℓ_{1}, ℓ_{2}).

The right hand side of (42) belongs to ℚ̄, by Part 1 of Proposition 1.12 and Theorem 1.13 with m = 0. Equation (42) should be interpreted to mean that the left hand side also belongs to ℚ̄, viewed as a subfield of ℂ_{p} under the chosen embeddings, and agrees with the right hand side. Note that although both sides of (42) depend on the choice of the differential ω_{A} that was made in the definition of the periods Ω(A) and Ω_{p}(A), the quantity ℒ_{p}(ν), just like its complex counterpart L_{𝔠}(ν^{−1}, 0), does not depend on this choice.

We are mainly interested in the behavior of ℒ_{p}(ν) at the so-called self-dual Hecke characters, which are defined as follows:

Definition 2.2. A Hecke character ν ∈ Σ_{crit}(𝔠) is said to be self-dual or anticyclotomic if

νν^{∗} = N_{K}.

Note that a self-dual character is necessarily of infinity type (1 + j, −j) for some j ∈ ℤ. There is also a restriction on the central character of such a ν. More precisely, it is clear that εν̄ = \overline{ε_{ν}}, while ε_{ν^{∗}} = ε_{ν}. If ν is a self-dual character, it follows that for any x ∈ 𝔸_{K}^{×},

ν(N_{ℚ}^{K}(x)) = ν(xx̄) = (νν^{∗})(x) = N_{K}(x) = N(N_{ℚ}^{K}(x)).

Figure 1. Critical infinity types for the Katz p-adic L-function

Hence

ν|_{N_{ℚ}^{K}𝔸_{K}^{×}} = N  and  ε_{ν}|_{N_{ℚ}^{K}𝔸_{K}^{×}} = 1.

This implies that the central character of a self-dual character is either 1 or ε_{K}, where ε_{K} denotes the quadratic Dirichlet character corresponding to the extension K/ℚ.

The reason for the terminology in Definition 2.2 is that the functional equation for the L-series L(ν^{−1}, s) relates L(ν^{−1}, s) to L(ν^{−1}, −s), and therefore s = 0 is the central critical point for this complex L-series. Also since the conductor of a self-dual character is clearly invariant under complex conjugation, we will assume henceforth that 𝔠 = 𝔠̄. Denote by Σ_{sd}(𝔠) the set of self-dual Hecke characters, and write

Σ_{sd}^{(1)}(𝔠) = Σ_{crit}^{(1)}(𝔠) ∩ Σ_{sd}(𝔠), Σ_{sd}^{(2)}(𝔠) = Σ_{crit}^{(2)}(𝔠) ∩ Σ_{sd}(𝔠).

In particular, the possible infinity types of characters in Σ_{sd}^{(2)}(𝔠) correspond to the black dots in Figure 1. The following is merely a restatement of Proposition 2.1 for self-dual characters.

Proposition 2.3. For all characters ν ∈ Σ_{sd}^{(2)}(𝔠) of infinity type (1 + j, −j) with j ≥ 0,

(43)  \frac{ℒ_{p}(ν)}{Ω_{p}(A)^{1+2j}} = (1 − ν^{−1}(𝔭̄))^{2} × \frac{j!(2π)^{j}L_{𝔠}(ν^{−1}, 0)}{\sqrt{D}^{j}Ω(A)^{1+2j}}.

The following theorem is the p-adic counterpart of Theorem 1.13.

Theorem 2.4. Suppose that ν ∈ Σ_{sd}^{(2)}(𝔠) is of infinity type (1 + j, −j), with j ≥ 0. Then

\frac{ℒ_{p}(ν)}{Ω_{p}(ν^{∗})} belongs to E_{ν}.

Proof. By definition of Ω_{p}(ν^{∗}),

\frac{ℒ_{p}(ν)}{Ω_{p}(ν^{∗})} = \frac{ℒ_{p}(ν)}{Ω_{p}(A)^{1+2j}} × \frac{(2πi)^{−j}Ω(A)^{1+2j}}{Ω(ν^{∗})}
= \frac{L(ν^{−1}, 0)}{(2πi)^{−j}Ω(A)^{1+2j}} × \frac{(2πi)^{−j}Ω(A)^{1+2j}}{Ω(ν^{∗})} = \frac{L(ν^{−1}, 0)}{Ω(ν^{∗})} (mod E_{ν}^{×}),

where the penultimate equality follows from the interpolation property of the Katz p-adic L-function in Proposition 2.3. The result is now a direct consequence of Theorem 1.13 with m = 0. □

Theorem 2.4 expresses ℒ_{p}(ν) as an E_{ν}-multiple of a p-adic period Ω_{p}(ν^{∗}), when ν lies in the range Σ_{sd}^{(2)}(𝔠) of classical interpolation for the Katz p-adic L-function. Our main interest is in obtaining analogous results for certain critical characters in Σ_{sd}^{(1)}(𝔠). These characters are outside the range of interpolation, and so (43) does not directly say anything about these values. Our approach to studying them relies on a different kind of p-adic L-function, namely one attached to Rankin-Selberg L-series, which we define and study in the following section.

2.2. p-adic Rankin L-series. In this section, we consider p-adic L-functions obtained by interpolating special values of Rankin-Selberg L-series associated to modular forms and Hecke characters of a quadratic imaginary field K of odd discriminant. We briefly recall the definition of this p-adic L-function that is given in Sec. 5 of [BDP], referring the reader to loc.cit. for a more detailed description.

Let S_{k}(Γ_{0}(N), ε) denote the space of cusp forms of weight k = r + 2 and character ε on Γ_{0}(N). Let f ∈ S_{k}(Γ_{0}(N), ε) be a normalized newform and let E_{f} denote the subfield of ℂ generated by its Fourier coefficients.

Definition 2.5. The pair (f, K) is said to satisfy the Heegner hypothesis if O_{K} contains a cyclic ideal of norm N, i.e., an integral ideal 𝔑 of O_{K} with O_{K}/𝔑 = ℤ/Nℤ.

Assume from now on that (f, K) satisfies the Heegner hypothesis, and let 𝔑 be a fixed cyclic O_{K}-ideal of norm N.

Definition 2.6. A Hecke character χ of K of infinity type (ℓ_{1}, ℓ_{2}) is said to be central critical for f if

ℓ_{1} + ℓ_{2} = k  and  ε_{χ} = ε.

The reason for the terminology of Definition 2.6 is that when χ satisfies these hypotheses, the complex Rankin L-series L(f, χ^{−1}, s) is self-dual and s = 0 is its central (critical) point.

We now pick a rational integer c prime to pN and denote by Σ_{cc}(c, 𝔑, ε) the set of central critical characters χ such that

𝔣_{χ} = 𝔣_{1}𝔣_{2}  with  𝔣_{1} | c, 𝔣_{2} | 𝔑.

The set Σ_{cc}(c, 𝔑, ε) can be expressed as a disjoint union

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

where Σ_{cc}^{(1)}(c, 𝔑, ε) and Σ_{cc}^{(2)}(c, 𝔑, ε) denote the subsets consisting of characters of infinity type (k + j, −j) with 1 − k ≤ j ≤ −1 and j ≥ 0 respectively. We shall also denote by Σ̂_{cc}(c, 𝔑, ε) the completion of Σ_{cc}(c, 𝔑, ε) relative to the p-adic compact open topology on Σ_{cc}(c, 𝔑, ε) which is defined in Section 5.4 of [BDP]. The infinity types of Hecke characters in Σ_{cc}^{(1)}(c, 𝔑, ε) and Σ_{cc}^{(2)}(c, 𝔑, ε) correspond respectively to the white and black dots in the shaded regions in Figure 2. We note that the set Σ_{cc}^{(2)}(c, 𝔑, ε) of classical central critical characters “of type 2” is dense in Σ̂_{cc}(c, 𝔑, ε).

For all χ ∈ Σ_{cc}^{(2)}(c, 𝔑, ε) of infinity type (k+j, −j) with j ≥ 0, let E_{f,χ} denote the subfield of ℂ generated by E_{f} and the values of χ, and let E_{f,χ,ε} be the field generated by E_{f,χ} and by the abelian extension of ℚ cut out by ε. The algebraic part of L(f, χ^{−1}, 0) is defined by the rule

(44)  L_{alg}(f, χ^{−1}, 0) := w(f, χ)^{−1}C(f, χ, c) · \frac{L(f, χ^{−1}, 0)}{Ω(A)^{2(k+2j)}} ,

Figure 2. Critical infinity types for the p-adic Rankin L-function

where w(f, χ)^{−1} ∈ E_{f,χ,ε} is the scalar (of complex norm 1) defined in equation (181) of [BDP] and C(f, χ, c) is the explicit real constant defined in Theorem 5.6 of [BDP]. For primitive χ, i.e. those such that c | 𝔣_{χ},

(45)  C(f, χ, c) = \frac{π^{r+2j+1}j!(r + 1 + j)!w_{K}}{\sqrt{D}^{r+1+2j}c^{r+3+2j}} ∏_{q|c}(q − ε_{K}(q))^{2},

where w_{K} = #O_{K}^{×} is the number of roots of unity in K. (Indeed, because χ is assumed to be primitive and D necessarily divides the conductor of ε_{f}, the set S(f) in the statement of Theorem 5.6. of [BDP] is empty in our setting.)

Theorem 6.5 of [BDP] shows that L_{alg}(f, χ^{−1}, 0) belongs to \overline{ℚ}. By analogy with the definition of the Katz p-adic L-function, it is natural to attempt to p-adically interpolate the special values L_{alg}(f, χ^{−1}, 0) as χ ranges over Σ_{cc}^{(2)}(c, 𝔑, ε).

Proposition 2.7. Let χ ↦ L_{p}(f, χ) be the function on Σ_{cc}^{(2)}(c, 𝔑, ε) defined by

(46)  L_{p}(f, χ) := Ω_{p}(A)^{2(k+2j)}(1 − χ^{−1}(𝔭̄)a_{p}(f) + χ^{−2}(𝔭̄)ε(p)p^{k−1})^{2}L_{alg}(f, χ^{−1}, 0),

for all χ of infinity type (k +j, −j) with j ≥ 0. This function extends (uniquely) to a p-adically continuous function on Σ̂_{cc}(c, 𝔑, ε).

This statement is proved in Proposition 6.10 of [BDP].

The function χ ↦ L_{p}(f, χ) on Σ̂_{cc}(c, 𝔑, ε) will be referred to as the p-adic Rankin L-function attached to the cusp form f.

2.3. A p-adic Gross-Zagier formula. In this section, we specialise to the case where the newform f is of weight k = 2, and assume that χ is a finite order Hecke character of K satisfying

χN_{K} belongs to Σ_{cc}^{(1)}(c, 𝔑, ε).

In particular, this character lies outside the domain Σ_{cc}^{(2)}(c, 𝔑, ε) of classical interpolation defining L_{p}(f, −). The p-adic Gross-Zagier formula alluded to in the title of this section relates the special value L_{p}(f, χN_{K}) to the formal group logarithm of a Heegner point on the modular abelian variety attached to f.

Recall that E_{f} and E_{χ} are the subfields of ℚ̄ generated, respectively, by the Fourier coefficients of f and the values of χ. Let E_{f,χ} denote the field generated by both E_{f} and E_{χ}, and let T_{f} ⊂ E_{f} and T_{f,χ} ⊂ E_{F,χ} denote their respective integer rings.

Let Γ := Γ_{ε}(N) ⊂ Γ_{0}(N) be the subgroup attached to f, defined by

(47)  Γ = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(N) such that ε(a) = 1.\} ,

The associated modular curve C has a model over ℚ obtained by realising C as the solution to a moduli problem, which we now describe. Given an abelian group G of exponent N, denote by G^{∗} the set of elements of G of order N. This set of “primitive elements” is equipped with a natural free action by (ℤ/Nℤ)^{×}, which is transitive when G is cyclic.

Definition 2.8. A Γ_{ε}(N)-level structure on an elliptic curve E is a pair (C_{N}, t), where (1) C_{N} is a cyclic subgroup scheme of E of order N,

(2) t is an orbit in C_{N}^{∗} for the action of ker ε. If E is an elliptic curve defined over a field L, then the Γ-level structure on E is defined over the field L if C_{N} is a group scheme over L and t is fixed by the natural action of Gal(L̄/L).

The curve C coarsely classifies the set of isomorphism classes of triples (E, C_{N}, t) where E is an elliptic curve and (C_{N}, t) is a Γ-level structure on E. When Γ is torsion-free (which occurs, for example, when ε is odd and N is divisible by a prime of the form 4n + 3 and a prime of the form 3n + 2) the curve C is even a fine moduli space; for any field L, one then has

C(L) = {Triples (E, C_{N}, t) defined over L}/L-isomorphism.

Since the datum of t determines the associated cyclic group C_{N}, we sometimes drop the latter from the notation, and write (E, t) instead of (E, C_{N}, t) when convenient. The group scheme A[𝔑] of 𝔑-torsion points in A is a cyclic subgroup scheme of A of order N defined over F. A Γ-level structure on A of the form (A[𝔑], t) is said to be of Heegner type (associated to the ideal 𝔑).

The Eichler-Shimura construction associates to f an abelian variety B_{f} with endomorphism by an order in T_{f}, and a surjective morphism

Φ_{f} : J_{1}(N)⟶B_{f}

of abelian varieties over ℚ, called the modular parametrisation, which is well-defined up to a rational isogeny and factors through the natural projection J_{1}(N)⟶J_{ε}(N). Let

ω_{f} = 2πif(τ)dτ ∈ Ω^{1}(J_{1}(N)/E_{f})

be the regular differential on J_{1}(N) attached to f, and let ω_{B_{f}} ∈ Ω^{1}(B_{f}/E_{f})^{T_{f}} be the unique regular differential satisfying

(48)  Φ_{f}^{∗}(ω_{B_{f}}) = ω_{f}.

Let A be an elliptic curve with endomorphisms by the order O_{c} = ℤ + cO_{K} of conductor c, defined over the ring class field H_{c} of conductor c. The finite order Hecke character χ can be viewed as a character

χ : Gal(H_{c,𝔑}/K)⟶E_{χ},

where H_{c,𝔑} is the finite abelian extension of the ring class field H_{c} generated by the 𝔑-torsion points of A. The pair (A, A[𝔑]) corresponds to a point on X_{0}(N)(H_{c}), and the triple (A, A[𝔑], t) (for any (Γ_{ε}(N)-level structure attached to A[𝔑]) corresponds to a point in X_{1}(N)(H_{c,𝔑}). Denote these points by [A, A[𝔑]] and [A, A[𝔑], t] respectively. Fix a cusp ∞ of X_{1}(N) which is defined over ℚ, and let

(49)  Δ = [A, A[𝔑], t] − (∞) ∈ J_{1}(N)(H_{c,𝔑}).

To the pair (f, χ) we associate the Heegner point by letting G = Gal(H_{c,𝔑}/K) and setting

(50)  P_{f}(χ) := ∑_{σ∈G} χ^{−1}(σ)Φ_{f}(Δ^{σ}) ∈ B_{f}(H_{c,𝔑}) ⊗_{T_{f}} E_{f,χ}.

The embedding of ℚ̄ into ℂ_{p} that was fixed from the outset allows us to consider the formal group logarithm

log_{ω_{B_{f}}} : B_{f}(H_{c,𝔑})⟶ℂ_{p}.

We extend this function to B_{f}(H_{c,𝔑}) ⊗_{T_{f}} E_{f,χ} by E_{f,χ}-linearity.

Theorem 2.9. With notations and assumptions as above,

L_{p}(f, χN_{K}) = (1 − χ^{−1}(𝔭̄)p^{−1}a_{p}(f) + χ^{−2}(𝔭̄)ε_{f}(p)p^{−1})^{2} log_{ω_{B_{f}}}^{2} (P_{f}(χ)).

Proof. Let

ℰ (f, χ) := (1 − χ^{−1}(𝔭̄)p^{−1}a_{p}(f) + χ^{−2}(𝔭̄)ε_{f}(p)p^{−1})^{2} ∈ E_{f,χ}^{×}

be the Euler factor appearing in the statement of Theorem 2.9. Let F denote the p-adic completion of H_{c,𝔑}. Theorem 6.13 of [BDP] in the case k = 2 and r = j = 0, with χ replaced by χN_{K}, gives

(51)  L_{p}(f, χN_{K}) = ℰ (f, χ) × (∑_{σ∈G} χ^{−1}(𝔞) · AJ_{F} (Δ^{σ})(ω_{f}))^{2} .

Note that in this context, the p-adic Abel-Jacobi map AJ_{F} that appears in (51) is related to the formal group logarithm by

AJ_{F} (Δ)(ω_{f}) = log_{ω_{f}} (Δ).

Therefore,

(52)  L_{p}(f, χN_{K}) = ℰ (f, χ) (∑_{σ∈G} χ^{−1}(σ) log_{ω_{f}} (Δ^{σ}))^{2} .

Theorem 2.9 follows from this formula and the fact that, by (48),

log_{ω_{f}} (Δ) = log_{Φ_{f}^{∗}(ω_{B_{f}})}(Δ) = log_{ω_{B_{f}}} (Φ_{f}(Δ)).
□

In the special case where f has rational Fourier coefficients and χ = 1 is the trivial character, the abelian variety B_{f} is an elliptic curve quotient of J_{0}(N) and the Heegner point P_{f} := P_{f}(1) belongs to B_{f}(K). Theorem 2.9 implies in this case that

(53)  L_{p}(f, N_{K}) = (\frac{p + 1 − a_{p}(f)}{p})^{2} log^{2}(P_{f}),

where log : B_{f}(K_{𝔭})⟶K_{𝔭} is the formal group logarithm attached to a rational differential on B_{f}/ℚ. Equation (53) exhibits a strong analogy with Theorem 1 of the Introduction, although it applies to arbitrary (modular) elliptic curves and not just elliptic curves with complex multiplication.

The remainder of Chapter 2 explains how Theorem 2.9 can in fact be used to prove Theorems 1 and 2 of the Introduction. The key to this proof is a relation between the Katz p-adic L-function of Section 2.1 and the p-adic Rankin L-function L_{p}(f, χ) of Section 2.2 in the special case where f is a theta series attached to a Hecke character of the imaginary quadratic field K. This explicit relation is described in the following section.

2.4. A factorisation of the p-adic Rankin L-series. This section focuses on the Rankin L-function L_{p}(f, χ) of f and K in the special case where f is a theta series associated to a Hecke character of the same imaginary quadratic field K.

More precisely, let ψ be a fixed Hecke character of K of infinity type (k − 1, 0) with k = r + 2 ≥ 2. Consider the associated theta series:

θ_{ψ} := ∑_{𝔞} ψ(𝔞)q^{N𝔞} = ∑_{n=1}^{∞} a_{n}(θ_{ψ})q^{n},

where the first sum is taken over integral ideals of K. The Fourier coefficients of θ_{ψ} generate a number field E_{θ_{ψ}} which is clearly contained in E_{ψ}.

The following classical proposition is due to Hecke and Schoenberg. (Cf. [Ogg] or Sec. 3.2 of [Za]).

Proposition 2.10. The theta series θ_{ψ} belongs to S_{k}(Γ_{0}(N), ε), where

(1) The level N is equal to DM, with M = N_{ℚ}^{K}𝔣_{ψ},

(2) The Nebentypus character ε is equal to ε_{K}ε_{ψ}.

Recall that an ideal of O_{K} is said to be cyclic if the quotient of O_{K} by this ideal is cyclic (as a group under addition).

Lemma 2.11. If the conductor 𝔣_{ψ} of ψ is a cyclic ideal of norm M prime to D, then θ_{ψ} satisfies the Heegner hypothesis relative to K.

Proof. In this case, the modular form θ_{ψ} is of level N = DM, by Proposition 2.10. But then the ideal

(54)  𝔑 := (\sqrt{−D})𝔣_{ψ}.

is a cyclic ideal of K of norm N. □

We will assume from now on that the condition in Lemma 2.11 is satisfied. Furthermore, we will always take 𝔑 to be the ideal in (54).

The goal of this section is to factor the p-adic Rankin L-function L_{p}(θ_{ψ}, χ) as a product of two Katz p-adic L-functions. As a preparation to stating the main result we record the following lemma:

Lemma 2.12. Let χ be any character in Σ_{cc}(c, 𝔑, ε).

(1) If χ belongs to Σ_{cc}^{(2)}(c, 𝔑, ε), then ψ^{−1}χ belongs to Σ_{sd}^{(2)}(c) and ψ^{∗−1}χ belong to Σ_{sd}^{(2)}(cN).
(2) If χ belongs to Σ_{cc}^{(1)}(c, 𝔑, ε), then ψ^{−1}χ belongs to Σ_{sd}^{(1)}(c) and ψ^{∗−1}χ belong to Σ_{sd}^{(2)}(cN).

Proof. This lemma follows from a direct verification which is left to the reader. It should be noted that when χ is of type (k + j, −j) then ψ^{−1}χ is of infinity type (1 + j, −j) and ψ^{∗−1}χ is of infinity type (1 + (1 + r + j), −(1 + r + j)). In particular, if χ lies on the edge of Σ_{cc}^{(1)}(c, 𝔑, ε), then ψ^{−1}χ lies on the edge of Σ_{sd}^{(1)}(c), i.e., is of infinity type (0, 1). □

Theorem 2.13. For all χ ∈ Σ_{cc}(c, 𝔑, ε),

(55)  L_{p}(θ_{ψ}, χ) = \frac{w(θ_{ψ}, χ)^{−1}w_{K}}{2^{r+1+2j}c^{r+3+2j}} ∏_{q|c}(q − ε_{K}(q))^{2} × ℒ_{p}(ψ^{−1}χ) × ℒ_{p}(ψ^{∗−1}χ).

Proof. Since Σ_{cc}^{(2)}(c, 𝔑, ε) is dense in Σ̂_{cc}^{(2)}(c, 𝔑, ε), it suffices to prove the formula for the characters χ in this range, where it follows directly from the interpolation properties defining the respective p-adic L-functions. More precisely, by (46),

(56)  \frac{L_{p}(θ_{ψ}, χ)}{Ω_{p}(A)^{2(k+2j)}} = ((1 − ψχ^{−1}(𝔭̄))(1 − ψ^{∗}χ^{−1}(𝔭̄))^{2} L_{alg}(θ_{ψ}, χ^{−1}, 0).

Let δ_{c} := ∏_{q|c}(q − ε_{K}(q))^{2}. By the definition of L_{alg}(θ_{ψ}, χ^{−1}, 0) given in (44) and (45),

L_{alg}(θ_{ψ}, χ^{−1}, 0) = w(θ_{ψ}, χ)^{−1}C(θ_{ψ}, χ, c)\frac{L(θ_{ψ}, χ^{−1}, 0)}{Ω(A)^{2(k+2j)}}
(57)  = w(θ_{ψ}, χ)^{−1}w_{K}δ_{c} \frac{π^{r+2j+1}j!(1 + r + j)!}{\sqrt{D}^{1+r+2j}c^{3+r+2j}} × \frac{L(ψχ^{−1}, 0)L(ψ^{∗}χ^{−1}, 0)}{Ω(A)^{2(k+2j)}}

= \frac{w(θ_{ψ}, χ)^{−1}w_{K}δ_{c}}{c^{r+3+2j}} (\frac{j!π^{j}L(ψχ^{−1}, 0)}{\sqrt{D}^{j}Ω(A)^{1+2j}}) × (\frac{(1 + r + j)!π^{1+r+j}L(ψ^{∗}χ^{−1}, 0)}{\sqrt{D}^{1+r+j}Ω(A)^{1+2(1+r+j)}}) .

Combining (56) and (57) with the interpolation property of the Katz p-adic L-function given in Proposition 2.3, we obtain

(58)  \frac{L_{p}(θ_{ψ}, χ)}{Ω_{p}(A)^{2(k+2j)}} = \frac{w(θ_{ψ}, χ)^{−1}w_{K}δ_{c}}{2^{r+1+2j}c^{r+3+2j}} × \frac{ℒ_{p}(ψ^{−1}χ)}{Ω_{p}(A)^{1+2j}} × \frac{ℒ_{p}(ψ^{∗−1}χ)}{Ω_{p}(A)^{1+2(1+r+j)}} .

Clearing the powers of Ω_{p}(A) on both sides gives the desired result. □

The Nebentypus character ε can be viewed as a finite order Galois character of G_{ℚ}. Recall that E_{ψ,χ,ε} denotes the smallest extension of E_{ψ,χ} containing the field through which this character factors.

Corollary 2.14. For all χ ∈ Σ_{cc}(c, 𝔑, ε), with 𝔠 = cN,

L_{p}(θ_{ψ}, χ) = ℒ_{p}(ψ^{−1}χ) × ℒ_{p}(ψ^{∗−1}χ) (mod E_{ψ,χ,ε}^{×}).

Proof. This follows from Theorem 2.13 in light of the fact that the constant that appears on the right hand side of (55) belongs to E_{ψ,χ,ε}^{×}. □

2.5. Proof of Rubin’s Theorem. The goal of this section is to prove Theorem 2 of the Introduction. Let ν ∈ Σ_{sd}(𝔠) be a Hecke character of infinity type (1, 0) satisfying conditions (6) and (7) of the Introduction.

Definition 2.15. A pair (ψ, χ) of Hecke characters is said to be good for ν if it satisfies the following conditions.

(1) The character ψ is of type (1, 0) and satisfies the Heegner hypothesis, so that the associated theta series θ_{ψ} belongs to S_{2}(Γ_{0}(N), ε) where (N) = 𝔑𝔑̄ is the norm of a cyclic ideal 𝔑 of O_{K} and ε is an appropriate even Dirichlet character of modulus N.

(2) The character χ is of finite order, and χN_{K} belongs to Σ_{cc}^{(1)}(c, 𝔑, ε).

(3) The character ψχ^{−1} is equal to ν, i.e., ψ^{−1}χN_{K} = ν^{∗}.

(4) The classical L-value L(ψ^{∗}χ^{−1}N_{K}^{−1}, 0) is non-zero, i.e., ℒ_{p}(ψ^{∗−1}χN_{K}) ≠ 0.

The modular abelian variety B_{θ_{ψ}} attached to ψ is a CM abelian variety in the sense of Definition 1.5. Hence it is K-isogenous to the CM abelian variety B_{ψ} constructed in Section 1.2. In particular, the modular parametrisation Φ_{ψ} := Φ_{θ_{ψ}} can be viewed as a surjective morphism of abelian varieties over K

(59)  Φ_{ψ} : J_{1}(N)⟶B_{ψ}.

Given a good pair (ψ, χ), recall the Heegner divisor Δ ∈ J_{1}(N)(H_{c,𝔑}) that was constructed in Section 2.3, and the Heegner point

(60)  P_{ψ}(χ) := P_{θ_{ψ}}(χ) = ∑_{σ∈G} χ^{−1}(σ)Φ_{ψ}(Δ^{σ}) ∈ B_{ψ}(H_{χ}) ⊗_{T_{ψ}} E_{ψ,χ}

that was defined in equation (50) of that section with f = θ_{ψ}. Recall also that ω_{ψ} is an E_{ψ}-vector space generator of Ω^{1}(B_{ψ}/E_{ψ})^{T_{ψ}}. Viewing the point P_{ψ}(χ) as a formal linear combination of elements of B_{ψ}(H_{χ}) with coefficients in E_{ψ,χ}, we define the expression log_{ω_{ψ}}(P_{ψ}(χ)) by E_{χ}-linearity.

In the rest of this section, we will denote by E′_{ψ,χ} the subfield of ℚ̄ generated by E_{ψ}, E_{χ}, and the abelian extension H′_{χ} of K cut out by the finite order characters χ and χ^{∗}. The motivation for singling out good pairs for a special definition lies in the following proposition.

Proposition 2.16. For any pair (ψ, χ) which is good for ν,

(61)  ℒ_{p}(ν^{∗}) = Ω_{p}(ψ^{∗}χ^{−1}) log_{ω_{ψ}}^{2}(P_{ψ}(χ)) (mod (E′_{ψ,χ})^{×}),

where Ω_{p}(ξ) is the p-adic period attached to a critical Hecke character ξ in Definition 1.14.

Proof. By Theorem 2.9 applied to f = θ_{ψ},

(62)  L_{p}(θ_{ψ}, χN_{K}) = log_{ω_{ψ}}^{2}(P_{ψ}(χ)) (mod E_{ψ,χ}^{×}).

On the other hand, since E′_{ψ,χ} contains E_{ψ,χ,ε}, Corollary 2.14 implies that

L_{p}(θ_{ψ}, χN_{K}) = ℒ_{p}(ψ^{−1}χN_{K})ℒ_{p}(ψ^{∗−1}χN_{K}) (mod (E′_{ψ,χ})^{×})
(63)  = ℒ_{p}(ν^{∗})ℒ_{p}(ψχ^{∗−1}) (mod (E′_{ψ,χ})^{×}),

where the second equality follows from condition 3 in the definition of a good pair. The character ψχ^{∗−1} lies in the range Σ_{sd}^{(2)}(𝔠) of classical interpolation for the Katz L-function, and ℒ_{p}(ψ^{∗−1}χN_{K}) = ℒ_{p}(νχ/χ^{∗}) is non-zero by condition 4 in the definition of a good pair. Therefore, by Theorem 2.4,

(64)  ℒ_{p}(ψχ^{∗−1}) = Ω_{p}(ψ^{∗}χ^{−1}) (mod E_{ψ,χ}^{×}).

Proposition 2.16 follows by combining the equalities in (62) and (63) and using (64). □

To go further, we will analyse the expressions occurring in the right hand side of (61) and relate them to quantities depending solely on ν and not on the good pair (ψ, χ).

Lemma 2.17. For all good pairs (ψ, χ) attached to ν,

Ω_{p}(ψ^{∗}χ^{−1}) = Ω_{p}(ν^{∗}) (mod (E′_{ψ,χ})^{×}).

Proof. Condition 3 in the definition of a good pair implies that ψ^{∗}χ^{−1} = ν^{∗}χ^{∗}/χ. The lemma therefore follows from Lemma 1.15, with ψ replaced by ν^{∗} and χ by the finite order character χ^{∗}/χ, which factors through Gal(H′_{χ}/K). □

It will be useful to view the point P_{ψ}(χ) appearing in (61) as an element of B_{ψ,χ}^{0}(H_{c,𝔑}) or as a K-rational point on the abelian variety B_{ψ,χ} that was introduced in Section 1.2. More precisely, after setting

(65)  P_{ψ}(χ) := ∑_{σ∈G} Φ_{ψ}(Δ^{σ}) ⊗ χ^{−1}(σ) ∈ B_{ψ}(K̄) ⊗_{T_{ψ}} T_{ψ,χ} = B_{ψ,χ}^{0}(K̄),

we observe that, for all τ ∈ Gal(K̄/K),

τ ∗_{0} P_{ψ}(χ) = ∑_{σ∈G} Φ_{ψ}(Δ^{τσ}) ⊗ χ^{−1}(σ)
= ∑_{σ∈G} Φ_{ψ}(Δ^{σ}) ⊗ χ^{−1}(στ^{−1}) = P_{ψ}(χ)χ(τ).

The point P_{ψ}(χ) therefore belongs to B_{ψ,χ}(K) by (33).

Recall the differentials ω_{ψ} and ω_{ψ,χ} ∈ Ω^{1}(B_{ψ,χ}/E_{ψ,χ})^{T_{ψ,χ}}.

Lemma 2.18. For all good pairs (ψ, χ) attached to ν = ψχ^{−1},

log_{ω_{ψ}}(P_{ψ}(χ)) = log_{ω_{ψ,χ}}(P_{ψ}(χ)).

Proof. Let G = Gal(H_{c,𝔑}/K) and let P = Φ_{ψ}(Δ). By definition,

log_{ω_{ψ}}(P_{ψ}(χ)) = ∑_{σ∈G} χ(σ)^{−1} log_{ω_{ψ}}(P^{σ}) = ∑_{σ∈G} χ(σ)^{−1} log_{i^{∗}(ω_{ψ,χ})}(P^{σ})
= ∑_{σ∈G} χ(σ)^{−1} log_{ω_{ψ,χ}}(P^{σ} ⊗ 1) = ∑_{σ∈G} log_{χ(σ)^{−1}ω_{ψ,χ}}(P^{σ} ⊗ 1)

= ∑_{σ∈G} log_{ω_{ψ,χ}}(P^{σ} ⊗ χ(σ)^{−1}) = log_{ω_{ψ,χ}} (∑_{σ∈G} P^{σ} ⊗ χ(σ)^{−1}) .

This completes the proof. □

Lemma 2.19. There exists P_{ν} ∈ B_{ν}(K) and ω_{ν} ∈ Ω^{1}(B_{ν}/E_{ν})^{T_{ν}} such that

log_{ω_{ψ,χ}}(P_{ψ}(χ)) = log_{ω_{ν}} (P_{ν}) (mod (E′_{ψ,χ})^{×}).

Proof. Recall from Lemma 1.10 that there is a K-rational isogeny

B_{ν} ⊗_{T_{ν}} T_{ψ,χ}⟶B_{ψ,χ}.

Composing it with the natural morphism B_{ν}⟶B_{ν} ⊗_{T_{ν}} T_{ψ,χ}, we obtain a T_{ν}-equivariant morphism j : B_{ν}⟶B_{ψ,χ} defined over K with finite kernel. The fact that L(ν, s) has a simple zero at s = 1 implies that B_{ν}(K) ⊗ ℚ is one-dimensional over E_{ν}, and therefore that B_{ψ,χ}(K) ⊗ ℚ is one-dimensional over E_{ψ,χ}. In particular, if P_{ν} is any generator of B_{ν}(K) ⊗ ℚ, we may write

P_{ψ}(χ) = λj(P_{ν})

for some non-zero scalar λ ∈ E_{ψ,χ}^{×}. But letting

ω_{ν} = j^{∗}(ω_{ψ,χ}) ∈ Ω^{1}(B_{ν}/E′_{ψ,χ})^{T_{ν}} ,

we have

log_{ω_{ψ,χ}}(P_{ψ}(χ)) = log_{ω_{ψ,χ}}(λj(P_{ν})) = log_{λ^{∗}ω_{ψ,χ}}(j(P_{ν})) = λ log_{ω_{ψ,χ}}(j(P_{ν}))
= λ log_{j^{∗}ω_{ψ,χ}}(P_{ν}) = λ log_{ω_{ν}} (P_{ν}).

The lemma now follows after multiplying ω_{ν} by an appropriate scalar in (E′_{ψ,χ})^{×} so that it belongs to Ω^{1}(B_{ν}/E_{ν})^{T_{ν}}. □

Proposition 2.20. There exists ω_{ν} ∈ Ω^{1}(B_{ν}/E_{ν})^{T_{ν}} and P_{ν} ∈ B_{ν}(K) such that

(66)  ℒ_{p}(ν^{∗}) = Ω_{p}(ν^{∗})^{−1} log_{ω_{ν}}^{2} (P_{ν}) (mod (E′_{ψ,χ})^{×}),

for all good pairs (ψ, χ) attached to ν.

Proof. This follows after invoking Lemmas 2.17, 2.18 and 2.19 to rewrite the expression appearing in the right hand side of Proposition 2.16. □

While Proposition 2.20 brings us close to Theorem 2 of the Introduction, it is somewhat more vague in that both sides of the purported equality may differ a priori by a non-zero element of the typically larger field E′_{ψ,χ}. The alert reader will also notice that this proposition is potentially vacuous for now, because the existence of a good pair for ν has not yet been established! The next proposition repairs this omission, and directly implies Theorem 2 of the Introduction.

Proposition 2.21. The set S_{ν} of pairs (ψ, χ) that are good for ν is non-empty. Furthermore,

(67)  ⋂_{(ψ,χ)∈S_{ν}} E′_{ψ,χ} = E_{ν}.

The proof of Proposition 2.21 rests crucially on a non-vanishing result of Rohrlich and Greenberg ([Ro], [Gre]) for the central critical values of Hecke L-series. In order to state it, we fix a rational prime ℓ which is split in K and let

K_{∞}^{−} = ∪_{n≥0}K_{n}^{−}

be the so-called anti-cyclotomic ℤ_{ℓ} extension of K; it is the unique ℤ_{ℓ}-extensions of K which is Galois over ℚ and for which Gal(K_{∞}^{−}/ℚ) = ℤ_{ℓ} ⋊ (ℤ/2ℤ) is a generalised dihedral group.

Lemma 2.22 (Greenberg, Rohrlich). Let ψ_{0} be a self-dual Hecke character of K of infinity type (1, 0). Assume that the sign w_{ψ_{0}} in the functional equation of L(ψ_{0}, s) is equal to 1. Then there are infinitely many finite-order characters χ of Gal(K_{∞}^{−}/K) for which L(ψ_{0}χ, 1) ≠ 0.

Proof. In light of the hypothesis that w_{ψ_{0}} = 1, Theorem 1 of [Gre] implies that the Katz p-adic Lfunction (with p = ℓ) does not vanish identically on any open ℓ-adic neighbourhood of ψ_{0} in Σ_{sd}(c), with c = cond(ψ_{0}), say. (Cf. the discussion in the first paragraph of the proof of Proposition 1 on p. 93 of [Gre].) If U is any sufficiently small such neighbourhood, then

(1) The restriction to U of the Katz p-adic L-function is described by a power series with p-adically bounded cofficients, and therefore admits only finitely many zeros by the Weierstrass preparation theorem.

(2) The region U contains contains a dense subset of points of the form ψ_{0}χ, where χ is a finite order character of Gal(K_{∞}^{−}/K).

Lemma 2.22 follows directly from these two facts. □

Proof of Proposition 2.21. Let S̄_{ν} ⊃ S_{ν} be the set of pairs satisfying conditions 1-3 in the definition of a good pair, but without necessarily requiring the more subtle fourth condition. The proof of Proposition 2.21 will be broken down into four steps.

Step 1. The set S̄_{ν} is non-empty.

To see this, let ψ be any Hecke character (of conductor 𝔪 prime to the conductor of ν and ν^{∗}, say) satisfying condition 1 in Definition 2.15. Setting χ = ψν^{∗}N_{K}^{−1}, the pair (ψ, χ) satisfies conditions 1 and 3 by construction. Furthermore, the character χN_{K} = ψν^{∗} is of type (1, 1) and its central character is equal to

ε_{χ} = ε_{ψ}ε_{ν^{∗}} = N^{2}ε_{M}ε_{K} = N^{2}ε,

where ε is the nebentype character attached to θ_{ψ}. It follows that the character χN_{K} belongs to Σ_{cc}(cN, 𝔑, ε) with 𝔑 = 𝔪 · \sqrt{−D}. (The integer c is related to the conductor of ν.) Therefore, the pair (ψ, χ) belongs to S̄_{ν}.

Step 2. Given (ψ, χ) ∈ S̄_{ν}, there exist (ψ_{1}, χ_{1}) and (ψ_{2}, χ_{2}) ∈ S_{ν} with E′_{ψ_{1},χ_{1}} ∩ E′_{ψ_{2},χ_{2}} ⊂ E′_{ψ,χ}.

To see this, let ℓ = λλ̄ be a rational prime which splits in K and is relatively prime to the class number of K and the conductors of ψ and χ. For such a prime, let

K_{∞} = ∪_{n≥0}K_{n}, K′_{∞} = ∪_{n≥0}K′_{n}

be the unique ℤ_{ℓ}-extensions of K which are unramified outside of λ and λ̄ respectively, with [K_{n} : K] = ℓ^{n} and likewise for K′_{n}. The condition that ℓ does not divide the class number of K implies that the fields K_{n} and K′_{n} are totally ramified at λ and λ̄ respectively. If α is any character of Gal(K_{∞}/K), the pair (ψ_{1}, χ_{1}) := (ψα, χα) still belongs to S̄_{ν}. (For instance, ψα satisfies the first condition in Definition 2.15, with 𝔑 replaced by 𝔑λ^{n} and N by Nℓ^{n}, for a suitable n ≥ 0.) Furthermore,

(68)  L(ψ_{1}^{∗}χ_{1}^{−1}N_{K}^{−1}, 0) = L(ψ^{∗}χ^{−1}N_{K}^{−1} · (α^{∗}/α), 0).

The character α^{∗}/α is an anticyclotomic character of K of ℓ-power order and conductor, and all such characters can be obtained by choosing α appropriately. The fact that (ψ, χ) is a good pair implies that the sign w_{ψ^{∗}χ^{−1}} is equal to 1. Hence, by Lemma 2.22, there exists a choice of α for which the L-value appearing on the right of (68) is non-vanishing. The corresponding pair (ψ_{1}, χ_{1}) belongs to S_{ν} and satisfies

E′_{ψ_{1},χ_{1}} ⊂ E′_{ψ,χ}ℚ(ζ_{ℓ^{n}})K_{n}K′_{n}

for some n. Repeating the same construction with a different rational prime in the place of ℓ yields a second pair (ψ_{2}, χ_{2}) ∈ S_{ν}. Since the extension E′_{ψ_{1},χ_{1}}/E_{ψ,χ} is totally ramified at the primes above ℓ, while E′_{ψ_{2},χ_{2}} is unramified at these primes, it follows that E′_{ψ_{1},χ_{1}} ∩ E′_{ψ_{2},χ_{2}} is contained in E′_{ψ,χ}, as was to be shown.

Thanks to Step 2, we are reduced to showing that

(69)  ⋂_{(ψ,χ)∈S̄_{ν}} E′_{ψ,χ} = E_{ν}.

The next step shows that the fields E′_{ψ,χ} can be replaced by E_{ψ,χ} in this equality.

Step 3. For all (ψ, χ) ∈ S̄_{ν}, there exists a finite order character α of G_{K} such that the pair (ψα, χα) belongs to S̄_{ν} and

E′_{ψ,χ} ∩ E′_{ψα,χα} ⊂ E_{ψ,χ}.

To see this, note that the finite order character χ has cyclic image, isomorphic to ℤ/nℤ say. Step 3 is completed by choosing a character α of order n in such a way that (ψα, χα) belongs to S̄_{ν} and

(H_{χ}H_{χ^{∗}}) ∩ (H_{χα}H_{χα^{∗}}) = K.

For instance, one could choose α to be totally ramified at a prime λ of norm a rational prime ℓ which is relatively prime the conductors of ψ and χ.

Step 4. We are now reduced to showing

(70)  ⋂_{(ψ,χ)∈S̄_{ν}} E_{ψ,χ} = E_{ν}.

We will do this by showing

(71)  There exists a pair (ψ, χ) ∈ S̄_{ν} such that E_{ψ,χ} = E_{ν}.

We begin by choosing an ideal 𝔪_{0} of O_{K} with the property that O_{K}/𝔪_{0} = ℤ/Mℤ is cyclic, and an odd quadratic Dirichlet character ε_{M} of conductor dividing M. Let ψ_{0} be any Hecke character satisfying

ψ_{0}((a)) = ε_{M}(a mod 𝔪_{0})a

on principal ideals (a) of K. Such a ψ_{0} satisfies condition 1 in Definition 2.15, and therefore, after letting χ_{0} be the finite order character satisfying

ν^{∗} = ψ_{0}^{−1}χ_{0}N_{K},

it follows that (ψ_{0}, χ_{0}) belongs to S̄_{ν}. Furthermore, the restriction of ψ_{0} to the group of principal ideals of K takes values in K, and therefore

(72)  χ_{0}(σ) ∈ E_{ν},  for all σ ∈ G_{H} := Gal(K̄/H).

The character ψ_{0} itself takes values in a CM field of degree [H : K], denoted E_{0}, which need not be contained in E_{ν} in general. To remedy this problem, let H_{0} be the abelian extension of the Hilbert class field H cut out by the character χ_{0}. Next, let H′_{0} be any abelian extension of K containing H such that

(1) There is an isomorphism u : Gal(H′_{0}/K)⟶Gal(H_{0}/K) of abstract groups such that the diagram

(73)  [figure: 0; Gal(H′_{0}/H); Gal(H′_{0}/K); Gal(H/K); 0; 0; Gal(H_{0}/H); Gal(H_{0}/K); Gal(H/K); 0],

commutes, where the dotted arrows indicate the isomorphisms induced by u and the other arrows are the canonical maps of Galois theory.

(2) The relative discriminant of H′_{0} over K is relatively prime to its conjugate (and therefore to the discriminant of K, in particular).

If the bottom exact sequence of groups in (73) is split, then the extension H′_{0} is readily produced, using class field theory. To handle the general case, we follow an approach that is suggested by the proof of Prop. 2.1.7 in [Se]. Let Φ̃ := Gal(H_{0}/K) and let ψ : G_{K}⟶Φ̃ be the homomorphism attached to the extension H_{0}. Since H is everywhere unramified over K, the restriction ψ_{v} of ψ to a decomposition group at the prime v of K maps the inertia subgroup I_{v} to ℭ := Gal(H_{0}/H). Let S be any finite set of primes of K which generates the class group of K and satisfies v ∈ S ⇒ v̄ ∉ S. We claim that there is a homomorphism ϵ : G_{K}⟶ℭ satisfying

ϵ_{v} = ψ_{v} on I_{v}, for all v ∉ S.

When the ground field K is replaced by ℚ and S = ∅, this is shown in Lemma 2.1.6 of [Se], and the proof given there adapts readily to our situation. The field H′_{0} can now be obtained as the fixed field of the kernel of the homomorphism ψϵ^{−1}. With the extension H′_{0} in hand, let α : Gal(H′_{0}/K)⟶E_{χ}^{×} be the finite order Hecke character given by

α(σ) = χ_{0}(u(σ))^{−1},

and set (ψ, χ) = (ψ_{0}α, χ_{0}α). By construction, (ψ, χ) belongs to S̄_{ν}. We claim that χ and ψ take values in E_{ν}. Since ν^{∗} = ψ^{−1}χN_{K}, it is enough to prove this statement for χ. But observe that, for all 𝔞 ∈ I_{𝔣} (where 𝔣 is divisible by the conductor of χ_{0}, χ, and ψ), we have

χ(𝔞) = χ_{0}(σ_{𝔞})/χ_{0}(u(σ_{𝔞})) = χ_{0}(σ_{𝔞}u(σ_{𝔞})^{−1}).

But the element σ_{𝔞}u(σ_{𝔞})^{−1} belongs to Gal(H_{0}/H) by construction, and hence χ_{0}(σ_{𝔞}^{−1}u(σ_{𝔞})) belongs to E_{ν} by (72). It follows that ψ and χ are E_{ν}–valued, and therefore E_{ψ,χ} = E_{ν}, as claimed in (71). □

2.6. Elliptic curves with complex multiplication. Theorem 2 of the Introduction admits an alternate formulation involving algebraic points on elliptic curves with complex multiplication rather than K-rational points on the CM abelian varieties B_{ν} of Theorem 1.6. The goal of this section is to describe this variant.

We begin by reviewing the explicit construction of B_{ν} in terms of CM elliptic curves. The reader is referred to §4 of [GS], whose treatment we largely follow, for a more detailed exposition.

Let F be any abelian extension of K for which

(74)  ν_{F} := ν ∘ N_{F/K}

becomes K-valued, and let 𝔣 be its conductor. The ideal 𝔣 is divisible by 𝔣_{ν}, and there exists an elliptic curve A/F with complex multiplication by O_{K} whose associated Grossencharacter is ν_{F}. (Cf. Thm. 6 of [Shi] and its corollary on p. 512.) Let

(75)  B := Res_{F/K}(A).

It is an abelian variety over K of dimension d := [F : K]. Let G := Gal(F/K) = hom_{K}(F, ℚ̄), where the natural identification between these two sets arises from the distinguished embedding of F into ℚ̄ that was fixed from the outset. By definition of the restriction of scalars functor, there are natural isomorphisms

B_{/F} = ∏_{σ∈G} A^{σ}, B(K̄) = A(K̄ ⊗_{K} F) = ∏_{σ∈G} A^{σ}(K̄)

of algebraic groups over F and abelian groups respectively. In particular, a point of B(K̄) is described by a d-tuple (P_{τ})_{τ∈G}, with P_{τ} ∈ A^{τ} (K̄). Relative to this identification, the Galois group G_{K} acts on B(K̄) on the left by the rule

(76)  ξ(P_{τ})_{τ} = (ξP_{τ})_{ξτ} , for all ξ ∈ G_{K}.

Consider the “twisted group ring”

(77)  T := ⊕_{σ∈G} Hom_{F} (A, A^{σ}) = {∑_{σ∈G} a_{σ}σ, with a_{σ} ∈ Hom_{F} (A, A^{σ})} ,

with multiplication given by

(78)  (a_{σ}σ)(a_{τ} τ) = a_{σ}a_{τ}^{σ}στ,

where the isogeny a_{τ}^{σ} belongs to hom_{F} (A^{σ}, A^{στ}) and the composition of isogenies in (78) is to be taken from left to right. The right action of T on B(K̄) defined by

(79)  (P_{τ})_{τ} ∗ (a_{σ}σ) := (a_{σ}^{τ}(P_{τ}))_{τσ}

commutes with the Galois action described in (76), and corresponds to a natural inclusion T ↪ End_{K}(B). The K-algebra E := T ⊗_{ℤ} ℚ is isomorphic to a finite product

E = ∏_{i} E_{i}

of CM fields, and dim_{K}(E) = dim(B). Therefore, the pair (B, E) is a CM abelian variety in the sense of Definition 1.5. The compatible system of ℓ-adic Galois representations attached to (B, E) corresponds to an E-valued algebraic Hecke character ν̃ in the sense of Definition 1.3, satisfying the relation

(80)  σ_{𝔞}(P) = P ∗ ν̃(𝔞), for all 𝔞 ∈ I_{𝔣ℓ} and P ∈ B(K̄)_{ℓ^{∞}},

where σ_{𝔞} ∈ G_{K}^{ab} denotes as before the Artin symbol attached to 𝔞 ∈ I_{𝔣ℓ}.

The element ν̃(𝔞) ∈ T is of the form φ_{𝔞}σ_{𝔞}, where

(81)  φ_{𝔞} : A → A^{σ_{𝔞}},

is an isogeny of degree N𝔞 satisfying

(82)  φ_{𝔞}(P) = P^{σ_{𝔞}},

for any P ∈ A[𝔤] with (𝔤, 𝔞) = 1. Note that the isogenies φ_{𝔞} satisfy the following cocycle condition:

(83)  φ_{𝔞𝔟} = φ_{𝔟}^{σ_{𝔞}} ∘ φ_{𝔞}.

The following proposition relates the Hecke characters ν̃ and ν.

Proposition 2.23. Given any homomorphism j ∈ Hom_{K}(E, ℂ), let ν_{j} := j ∘ ν̃ be the corresponding ℂ-valued Hecke character of K of infinity type (1, 0). The assignment j ↦ ν_{j} gives a bijection from Hom_{K}(E, ℂ) to the set Σ_{ν,F} of Hecke characters ν′ of K (of infinity type (1, 0)) satisfying

ν′ ∘ N_{F/K} = ν ∘ N_{F/K}.

Proposition 2.23 implies that there is a unique homomorphism j_{ν} ∈ Hom_{K}(E, ℂ) satisfying j_{ν} ∘ ν̃ = ν. In particular, j_{ν} maps E to E_{ν} and T to a finite index subring of T_{ν}. The abelian variety B_{ν} attached to ν in Theorem 1.6 can now be defined as the quotient B ⊗_{T,j_{ν}} T_{ν}. In subsequent constructions, it turns out to be more useful to realise B_{ν} as a subvariety of B, which can be done by setting

(84)  B_{ν} := B[ker j_{ν}].

The natural action of T on B_{ν} factors through the quotient T/ ker(j_{ν}), an integral domain having E_{ν} as field of fractions.

Consider the inclusion

(85)  i_{ν} : B_{ν}(K) ↪ B(K) = A(F),

where the last identification arises from the functorial property of the restriction of scalars. The following Proposition gives an explicit description of the image of (B_{ν}(K) ⊗ E_{ν})^{T_{ν}} in A(F) ⊗_{O_{K}} E_{ν} under the inclusion i_{ν} obtained from (85).

Proposition 2.24. Let Ẽ be any field containing E_{ν}. The inclusion i_{ν} of (85) identifies (B_{ν}(K) ⊗ Ẽ)^{T_{ν}} with

(A(F) ⊗_{O_{K}} Ẽ)^{ν} := {P ∈ A(F) ⊗_{O_{K}} Ẽ such that φ_{𝔞}(P) = ν(𝔞) × P^{σ_{𝔞}}, for all 𝔞 ∈ I_{𝔣}} .

Proof. It follows from the definitions that i_{ν}(B_{ν}(K)) is identified with the set of (P_{τ}) with P_{τ} ∈ A^{τ} (K̄) satisfying

(86)  ξP_{τ} = P_{ξτ} , for all ξ ∈ G_{K}.

Furthermore, if such a (P_{τ}) belongs to (B_{ν}(K) ⊗ E_{ν})^{T_{ν}}, then after setting ν̃(𝔞) = φ_{𝔞}σ_{𝔞} as in (81), we also have

(87)  (φ_{𝔞}^{τ}(P_{τ}))_{τσ_{𝔞}} = (P_{τ})_{τ} ∗ ν̃(𝔞) = (ν(𝔞)P_{τ})_{τ} .

Equating the σ_{𝔞}-components of these two vectors gives

φ_{𝔞}(P_{1}) = ν(𝔞)P_{σ_{𝔞}} = ν(𝔞)σ_{𝔞}P_{1},

where 1 is the identity embedding of F and the last equality follows from (86). The Proposition follows directly from this, after noting that the identification of B(K) with A(F) is simply the one sending (P_{τ})_{τ} to P_{1}. □

Given a global field F as in (74), let F_{ν} denote the subfield of ℚ̄ generated by F and E_{ν}. Recall that ω_{A} ∈ Ω^{1}(A/F) is a non-zero regular differential and that Ω_{p}(A) is the associated p-adic period.

Theorem 2.25. There exists a point P_{A,ν} ∈ (A(F) ⊗_{O_{K}} E_{ν})^{ν} such that

ℒ_{p}(ν^{∗}) = Ω_{p}(A)^{−1} log_{ω_{A}}^{2}(P_{A,ν}) (mod F_{ν}^{×}).

The point P_{A,ν} is non-zero if and only if L′(ν, 1) ≠ 0.

Proof. Theorem 2 of the Introduction asserts that

(88)  ℒ_{p}(ν^{∗}) = Ω_{p}(ν^{∗})^{−1} log_{ω_{ν}}^{2} (P_{ν}),

for some point P_{ν} ∈ B_{ν}(K) ⊗ ℚ which is non-trivial if and only if L′(ν, 1) ≠ 0. Invoking Lemma 1.15, we find

(89)  Ω_{p}(ν^{∗})^{−1} = Ω_{p}(A)^{−1} (mod F_{ν}^{×}).

Furthermore, by 2.24, we can view P_{ν} as a point P_{A,ν} ∈ (A(F) ⊗_{O_{K}} E_{ν})^{ν}, and we have

(90)  log_{ω_{ν}} (P_{ν}) = log_{ω_{A}}(P_{A,ν}) (mod F_{ν}^{×}).

Theorem 2.28 now follows by rewriting (88) using (89) and (90). □

2.7. A special case. This section is devoted to a more detailed and precise treatment of Theorem 2.25 under the following special assumptions:

(1) The quadratic imaginary field K has class number one, odd discriminant, and unit group of order two. This implies that K = ℚ(\sqrt{−D}) where D := − Disc(K) belongs to the finite set

S := {7, 11, 19, 43, 67, 163}.

(2) Let ψ_{0} be the Hecke character of K of infinity type (1, 0) given by the formula

(91)  ψ_{0}((a)) = ε_{K}(a mod \sqrt{−D})a.

The character ψ_{0} determines (uniquely, up to an isogeny) an elliptic curve A/ℚ satisfying

End_{K}(A) = O_{K}, L(A/ℚ, s) = L(ψ_{0}, s).

After fixing A, we will also write ψ_{A} instead of ψ_{0}. It can be checked that the conductor of ψ_{A} is equal to \sqrt{−D}, and that

ψ_{A}^{∗} = ψ̄_{A},  ψ_{A}ψ_{A}^{∗} = N_{K},  ε_{ψ_{A}} = ε_{K}.

Remark 2.26. The rather stringent assumptions on K that we have imposed exclude the arithmetically interesting, but somewhat idiosyncratic, cases where K = ℚ(\sqrt{−3}), ℚ(i), and ℚ(\sqrt{−2}). The detailed study of Chow-Heegner points in these special cases is the subject of Dong Quan Nguyen Ngoc’s forthcoming PhD thesis.

With the above assumptions, the character ψ_{A} can be used to give an explicit description of the set Σ_{sd}(c\sqrt{−D}):

Lemma 2.27. Let c be an integer prime to D, and let ν be a Hecke character in Σ_{sd}(c\sqrt{−D}). Then ν is of the form

ν = ψ_{A}χ^{−1},

where χ is a finite order ring class character of K of conductor dividing c.

Proof. The fact that ν and ψ_{A} both have central character ε_{K} implies that χ is a ring class character. Its conductor divides c\sqrt{−D}, since the same is true for ν and ψ_{A}. But it is also clear that χ is unramified at \sqrt{−D}, since the local components of ν and ψ_{A} at this prime of K are equal. The result follows. □

Given a ring class character χ of conductor c as above with values in a field E_{χ}, let

(92)  (A(H_{c}) ⊗_{O_{K}} E_{χ})^{χ} := {P ∈ A(H_{c}) ⊗_{O_{K}} E_{χ} such that σP = χ(σ)P, ∀σ ∈ Gal(H_{c}/K)}.

Finally, choose a regular differential ω_{A} ∈ Ω^{1}(A/K), and write Ω_{p}(A) for the p-adic period attached to this choice as in Section 2.1. Since A = B_{ψ} is the abelian variety attached to ψ, it follows that Ω_{p}(ψ_{A}^{∗}) = Ω_{p}(A).

The following theorem is a more precise variant of Theorem 2.25.

Theorem 2.28. Let χ be a ring class character of K of conductor prime to \sqrt{−D}. Then there exists a point P_{A}(χ) ∈ (A(H_{χ}) ⊗_{O_{K}} E_{χ})^{χ} such that

ℒ_{p}(ψ_{A}^{∗}χ) = Ω_{p}(A)^{−1}𝔤(χ) log_{ω_{A}}^{2}(P_{A}(χ)) (mod E_{χ}^{×}).

The point P_{A}(χ) is non-zero if and only if L′(ψ_{A}χ^{−1}, 1) ≠ 0.

Proof. By Theorem 2 of the Introduction,

(93)  ℒ_{p}(ψ_{A}^{∗}χ) = ℒ_{p}(ν^{∗}) = Ω_{p}(ν^{∗})^{−1} log_{ω_{ν}}^{2} (P_{ν}),

for some point P_{ν} ∈ B_{ν}(K) ⊗ ℚ which is non-trivial if and only if L′(ψ_{A}χ^{−1}, 1) ≠ 0. Using the fact that χ^{∗−1} = χ and invoking Lemma 1.15, we find

(94)  Ω_{p}(ν^{∗})^{−1} = Ω_{p}(ψ_{A}^{∗}χ^{∗−1})^{−1} = Ω_{p}(A)^{−1}𝔤(χ)^{−1} (mod E_{χ}^{×}).

After noting that (as in equation (31)) B_{ν} = B_{ψ,χ} = (A ⊗_{O_{K}} T_{χ})^{χ^{−1}} as abelian varieties over K, we observe that ω_{ν} = ω_{ψ,χ} and that the point P_{ν} ∈ B_{ν}(K) can be written as

P_{ν} = ∑_{σ∈G} P^{σ} ⊗ χ^{−1}(σ),

for some P ∈ A(H_{c}) ⊗ ℚ. Letting P_{A,χ} be the corresponding element in A(H_{c}) ⊗_{O_{K}} E_{χ} given by

P_{A,χ} = ∑_{σ∈G} χ^{−1}(σ)P^{σ},

we have

(95)  log_{ω_{ν}}^{2} (P_{ν}) = log_{ω_{ψ,χ}}^{2}(P_{ν}) = 𝔤(χ)^{2} log_{ω_{ψ,χ}^{0}}^{2}(P_{ν}) = 𝔤(χ)^{2} log_{ω_{A}}^{2}(P_{A,χ}) (mod E_{χ}^{×}),

where the second equality follows from Lemma 1.9 and the last from Lemma 2.18. Theorem 2.28 now follows by rewriting (93) using (94) and (95). □

In the special case where χ is a quadratic ring class character of K, cutting out an extension L = K(\sqrt{a}) of K, we obtain

(96)  ℒ_{p}(ψ_{A}^{∗}χ) = Ω_{p}(A)^{−1}\sqrt{a} log_{ω_{A}}^{2}(P_{A,L}^{−}) (mod K^{×}),

where P_{A,L}^{−} is a K-vector space generator of the trace 0 elements in A(L) ⊗ ℚ. Since in this case ψ_{A}χ is the Hecke character attached to a CM elliptic curve over ℚ, one recovers from (96) Rubin’s Theorem 1 of the Introduction.

3. Chow-Heegner points

The goal of the first three sections of this chapter is to recall the construction of the motives attached to Hecke characters and to modular forms. The remaining three sections are devoted to the definition of Chow-Heegner points on CM elliptic curves, as the image of generalised Heegner cycles by modular parametrisations attached to CM forms.

3.1. Motives for rational and homological equivalence. We begin by laying down our conventions regarding motives, following [Del]. We will work with either Chow motives or Grothendieck motives. For X a nonsingular variety over a number field F, let C^{m}(X) denote the group of algebraic cycles of codimension m on X defined over F. Let ∼ denote rational equivalence in C^{m}(X), and set

C^{m}(X) := C^{m}(X)/ ∼ .

Given two nonsingular varieties X and Y over F, and E any number field, we define the groups of correspondences

Corr^{m}(X, Y) := C^{dim X+m}(X × Y) Corr^{m}(X, Y)_{E} := Corr^{m}(X, Y) ⊗_{ℤ} E.

Definition 3.1. A motive over F with coefficients in E is a triple (X, e, m) where X/F is a nonsingular projective variety, e ∈ Corr^{0}(X, X)_{E} is an idempotent, and m is an integer.

Definition 3.2. The category M_{F,E} of Chow motives is the category whose objects are motives over F with coefficients in E, with morphisms defined by

Hom_{M_{F,E}}((X, e, m), (Y, f, n)) = f ∘ Corr^{n−m}(X, Y)_{ℚ} ∘ e.

The category M_{F,E}^{hom} of Grothendieck motives is defined in exactly the same way, but with homological equivalence replacing rational equivalence. We will denote the corresponding groups of cycle classes by C_{0}^{r}(X), Corr_{0}^{m}(X, Y), Corr_{0}^{m}(X, Y)_{E} etc.

Since rational equivalence is finer than homological equivalence, there is a natural functor

M_{F,E} → M_{F,E}^{hom},

so that every Chow motive gives rise to a Grothendieck motive. Further, the category of Grothendieck motives is equipped with natural realisation functors arising from any cohomology theory satisfying the Weil axioms. We now recall the description of the image of a motive M = (X, e, m) over F with coefficients in E under the most important realizations:

The Betti realisation: Recall that our conventions about number fields supply us with an embedding F ⟶ℂ. The Betti realisation is defined in terms of this embedding by

M_{B} := e · (H^{∗}(X(ℂ), ℚ)(m) ⊗ E).

It is a finite-dimensional E-vector space with a natural E-Hodge structure arising from the comparison isomorphism between the singular cohomology and the de Rham cohomology over ℂ.

The ℓ-adic realisation: Let X̄ denote the extension of X to ℚ̄. The ℓ-adic cohomology of X̄ gives rise to the ℓ-adic étale realisation of M:

M_{ℓ} := e · (H_{et}^{∗}(X̄, ℚ_{ℓ}(m)) ⊗ E) .

It is a free E ⊗ ℚ_{ℓ}-module of finite rank equipped with a continuous linear G_{F} -action.

The de Rham realisation: The de Rham realisation of M is defined by

M_{dR} := e · (H_{dR}^{∗}(X/F)(m) ⊗_{ℚ} E),

where H_{dR}^{∗}(X/F) denotes the hypercohomology of the deRham complex of sheaves over X. The module M_{dR} is a free E ⊗ F -module of finite rank equipped with a decreasing, separated and exhaustive Hodge filtration.

Moreover, there are natural comparison isomorphisms

(97)  M_{B} ⊗_{ℚ} ℂ ≃ M_{dR} ⊗_{F} ℂ,
(98)  M_{B} ⊗_{ℚ} ℚ_{ℓ} ≃ M_{ℓ},

which are E ⊗ ℂ-linear and E ⊗ ℚ_{ℓ}-linear respectively. Thus

rank_{E} M_{B} = rank_{E⊗F} M_{dR} = rank_{E⊗ℚ_{ℓ}}(M_{ℓ}),

and this common integer is called the E-rank of the motive M.

We define the period Ω(M) to be the determinant of the comparison isomorphism (97) relative to any choice of E-basis for M_{B} and EF basis of M_{dR} ⊗_{F} EF. Thus the period Ω(M) is a well-defined element of

ℂ^{×}/(EF)^{×}.

Remark 3.3. If F is a p-adic field, one also has a comparison isomorphism

(99)  M_{p} ⊗_{ℚ_{p}} B_{dR,p} ≃ M_{dR} ⊗_{F} B_{dR,p},

where B_{dR,p} is Fontaine’s ring of p-adic periods, which is endowed with a decreasing, exhaustive filtration and a continuous G_{F} -action. This comparison isomorphism is compatible with natural filtrations and G_{F} -actions on both sides.

Remark 3.4. Our definition of motives with coefficients coincides with Language B of Deligne [Del]. There is an equivalent way of defining motives with coefficients (the Language A) where the objects are motives M in M_{F,ℚ} equipped with the structure of an E-module: E → End(M), and morphisms are those that commute with the E-action. We refer the reader to Sec. 2.1. of loc. cit. for the translation between these points of view.

3.2. The motive of a Hecke character. For more general algebraic Hecke characters which are not of type (1, 0), one no longer has an associated abelian variety. Nevertheless, such a character still gives rise to a motive over K with coefficients in the field generated by its values. In the following two sections, we will quickly recall some facts about the motive of a Hecke character ψ of K.

By taking Tate twists and duals, it suffices to construct this motive when ψ has infinity type (r, 0), with r a positive integer. So let us suppose that ψ : 𝔸_{K}^{×} → ℂ^{×} is such a Hecke character of conductor 𝔣_{ψ}, and let E_{ψ} be the field generated over K by the values of ψ on the finite idèles. Pick an abelian extension F of K containing the ray class field K(𝔣_{ψ}) such that ψ_{F} := ψ ∘ N_{F/K} satisfies the equation

ψ_{F} = ψ_{A}^{r}.

Here ψ_{A} is the Hecke character of F with values in K associated to an elliptic curve A/F with complex multiplication by O_{K}.

We construct the motive M(ψ_{F}) ∈ M_{F,K} associated to ψ_{F} by considering an appropriate piece of the middle cohomology of the variety A^{r} over F. As in the Introduction, write [α] for the element of End_{F} (A) ⊗_{ℤ} ℚ corresponding to an element α ∈ K. Define an idempotent e_{r} ∈ Corr^{0}(A^{r}, A^{r})_{ℚ} by

e_{r} := (\frac{\sqrt{−D} + [\sqrt{−D}]}{2\sqrt{−D}})^{⊗r} + (\frac{\sqrt{−D} − [\sqrt{−D}]}{2\sqrt{−D}})^{⊗r} .

Then M(ψ_{F}) is the motive in M_{F,K} defined (in language A) by

M(ψ_{F}) := (A^{r}, e_{r}, 0),

where K acts diagonally on A^{r}. The ℓ-adic étale realisation M(ψ_{F})_{ℓ} of this motive is free of rank one over K ⊗ ℚ_{ℓ}, and G_{F} acts on it via ψ_{F}, viewed as a (K ⊗ ℚ_{ℓ})^{×}-valued Galois character:

M(ψ_{F})_{ℓ} = e_{r}H_{et}^{r}(\overline{A^{r}}, ℚ_{ℓ}) = (K ⊗ ℚ_{ℓ})(ψ_{F}).

The de Rham realisation M(ψ_{F})_{dR} is a free one-dimensional F ⊗_{ℚ}K-vector space, generated as an F -vector space by the classes of

ω_{A}^{r} := e_{r}(ω_{A} ∧ ⋯ ∧ ω_{A})  and  η_{A}^{r} := e_{r}(η_{A} ∧ ⋯ ∧ η_{A}),

where η_{A} is the unique class in H_{dR}^{1}(A/F) satisfying

[α]^{∗}η_{A} = ᾱη_{A} for all α ∈ K,  and  ⟨ω_{A}, η_{A}⟩ = 1.

The Hodge filtration on M(ψ_{F})_{dR} is given by

Fil^{0} M(ψ_{F})_{dR} = M(ψ_{F})_{dR} = F · ω_{A}^{r} + F · η_{A}^{r},
Fil^{1} M(ψ_{F})_{dR} = ⋯ = Fil^{r} M(ψ_{F})_{dR} = F · ω_{A}^{r},
Fil^{r+1} M(ψ_{F})_{dR} = 0.

It can be shown that M(ψ_{F}) descends to a motive M(ψ) ∈ M_{K,E_{ψ}}, whose ℓ-adic realisation is a free rank one module over Eψ ⊗ ℚ_{ℓ} on which G_{K} acts via the character ψ.

3.3. Deligne-Scholl motives. In this section, we will let ψ be a Hecke character of K of infinity type (r + 1, 0). This Hecke character gives rise to a theta-series

θ_{ψ} = ∑_{n=1}^{∞} a_{n}(θ_{ψ})q^{n} ∈ S_{r+2}(Γ_{0}(N), ε),

as in Proposition 2.10. Observe that the subfield of ℚ̄ generated by the Fourier coefficients a_{n}(θ_{ψ}) is always contained (albeit sometimes properly) in E_{ψ}. Henceforth, we will view θ_{ψ} as a modular form with Fourier coefficients in E_{ψ}.

Deligne has attached to θ_{ψ} a compatible system {V_{ℓ}(θ_{ψ})} of two-dimensional ℓ-adic representations of G_{ℚ} with coefficients in E_{ψ} ⊗ ℚ_{ℓ}, such that for any prime p ∤ Nℓ, the characteristic polynomial of the Frobenius element at p is given by

X^{2} − a_{p}(θ_{ψ})X + ε(p)p^{r+1}.

This representation is realised in the middle ℓ-adic cohomology of a variety which is fibered over a modular curve. More precisely, recall the group Γ = Γ_{ε}(N) defined in equation (47) and the modular curve C whose complex points are identified with Γ\H^{∗}. Let W_{r} be the r-th Kuga-Sato variety over C. It is a canonical compactification and desingularisation of the r-fold self-product of the universal elliptic curve over C. (See for example [BDP], Chapter 2 for more details on this definition.)

Theorem 3.5. (Scholl) There is a projector e_{θ_{ψ}} ∈ Corr^{0}(W_{r}, W_{r}) ⊗ E_{ψ} whose associated Grothendieck motive M(θ_{ψ}) := (W_{r}, e_{θ_{ψ}}, 0) satisfies

M(θ_{ψ})_{ℓ} ≃ V_{ℓ}(θ_{ψ})

as E_{ψ}[G_{ℚ}]-modules.

We remark that M(θ_{ψ}) is a motive over ℚ with coefficients in E_{ψ}, and that its ℓ-adic realisation M(θ_{ψ})_{ℓ} is identified with e_{θ_{ψ}}(H_{et}^{r+1}(W̄_{r}, ℚ_{ℓ}) ⊗_{ℚ} E_{ψ}).

The de Rham realisation

M(θ_{ψ})_{dR} = e_{θ_{ψ}}H_{dR}^{r+1}(W_{r}/E_{ψ})

is a two-dimensional E_{ψ}-vector space equipped with a canonical decreasing, exhaustive and separated Hodge filtration. This vector space and its associated filtration can be described concretely in terms of the cusp form θ_{ψ} as follows.

Let C^{0} denote the complement in C of the subscheme formed by the cusps. Setting W_{r}^{0} := W_{r} ×_{C} C^{0}, there is a natural analytic uniformization

W_{r}^{0}(ℂ) = (ℤ^{2r} ⋊ Γ)\(ℂ^{r} × H),

where the action of ℤ^{2r} on ℂ^{r} × H is given by

(100)  (m_{1}, n_{1}, …, m_{r}, n_{r})(w_{1}, …, w_{r}, τ) := (w_{1} + m_{1} + n_{1}τ, …, w_{r} + m_{r} + n_{r}τ, τ),

and Γ acts by the rule

(101)  ( \begin{matrix}a & b \\ c & d \end{matrix} ) (w_{1}, …, w_{r}, τ) = (\frac{w_{1}}{cτ + d}, …, \frac{w_{r}}{cτ + d}, \frac{aτ + b}{cτ + d}) .

The holomorphic (r+1) form

(102)  ω_{θ_{ψ}} := (2πi)^{r+1}θ_{ψ}(τ)dw_{1} ⋯ dw_{r}dτ

on W_{r}^{0}(ℂ) extends to regular differentials on W_{r}. This differential is defined over the field E_{ψ}, by the q-expansion principle, hence lies in H_{dR}^{r+1}(W_{r}) ⊗_{ℚ} E_{ψ}. Its class generates the (r + 1)-st step in the Hodge filtration:

(103)  Fil^{r+1}(M(θ_{ψ})_{dR}) = e_{θ_{ψ}} Fil^{r+1} H_{dR}^{r+1}(W_{r}) = E_{ψ} · ω_{θ_{ψ}}.

3.4. Modular parametrisations attached to CM forms. In this section, we will explain how the Tate conjectures imply the existence of algebraic cycle classes generalising those in Conjecture 7 of the Introduction. We first recall the statement of the Tate conjecture.

Conjecture 3.6 (Tate). Let V be a smooth projective variety over a number field F. Then the ℓ-adic étale cycle class map

(104)  cl_{ℓ} : CH^{j}(V)(F) ⊗ ℚ_{ℓ}⟶H_{et}^{2j}(V̄ , ℚ_{ℓ})(j)^{G_{F}}

is surjective.

A class in the target of (104) is called an ℓ-adic Tate cycle. The Tate conjecture will be used in our constructions through the following simple consequence. Recall the Chow groups CH^{d}(V)(F) defined in the Introduction.

Lemma 3.7. Let V_{1} and V_{2} be smooth projective varieties of dimension d over a number field F, and let e_{j} ∈ Corr^{0}(V_{j}, V_{j}) ⊗ E (for j = 1, 2) be idempotents satisfying

e_{j}H_{et}^{∗}(V̄_{j}, ℚ_{ℓ}) ⊗ E = e_{j}H_{et}^{d}(V̄_{j}, ℚ_{ℓ}) ⊗ E, j = 1, 2.

Let M_{j} := (V_{j}, e_{j}, 0), be the associated motives over F with coefficients in E, and suppose that the ℓ-adic realisations of M_{1} and M_{2} are isomorphic as (E ⊗ ℚ_{ℓ})[G_{F}]-modules. If Conjecture 3.6 is true for V_{1} × V_{2}, then there exists a correspondence Π ∈ CH^{d}(V_{1} × V_{2})(F) ⊗ E for which

(1) the induced morphism

(105)  Π_{ℓ} : (M_{1})_{ℓ}⟶(M_{2})_{ℓ}

of ℓ-adic realisations is an isomorphism of E ⊗ ℚ_{ℓ}[G_{F}]-modules;

(2) the induced morphism

(106)  Π_{dR} : (M_{1})_{dR}⟶(M_{2})_{dR}

is an isomorphism of E ⊗ F -vector spaces.

Proof. Let

h : e_{1}H_{et}^{d}(V̄_{1}, E ⊗ ℚ_{ℓ}) ≃ e_{2}H_{et}^{d}(V̄_{2}, E ⊗ ℚ_{ℓ})

be any isomorphism of (E ⊗ ℚ_{ℓ})[G_{F}]-modules. It corresponds to a Tate cycle

Z_{f} ∈ (H_{et}^{d}(V̄_{1}, E ⊗ ℚ_{ℓ})^{∨} ⊗ H_{et}^{d}(V̄_{2}, E ⊗ ℚ_{ℓ}))^{G_{F}}
= (H_{et}^{d}(V̄_{1}, E ⊗ ℚ_{ℓ}(d)) ⊗ H_{et}^{d}(V̄_{2}, E ⊗ ℚ_{ℓ}))^{G_{F}}
⊂ (H_{et}^{2d}(\overline{V_{1} × V_{2}}, E ⊗ ℚ_{ℓ}(d)))^{G_{F}} ,

where the superscript ^{∨} in the first line denotes the E-linear dual, the second line follows from the Poincaré duality, and the third from the Künneth formula. By Conjecture 3.6, there are elements α_{1}, …, α_{t} ∈ E⊗ℚ_{ℓ} and cycles Π_{1}, …, Π_{t} ∈ CH^{d}(V_{1} × V_{2})(F) satisfying

Z_{f} = ∑_{j=1}^{t} α_{j} cl_{ℓ}(Π_{j}).

After multiplying Z_{f} by a suitable power of ℓ, we may assume without loss of generality that the coefficients α_{j} belong to O_{E} ⊗ ℤ_{ℓ}. If (β_{1}, …, β_{t}) ∈ O_{E}^{t} is any vector which is sufficiently close to (α_{1}, …, α_{t}) in the ℓ-adic topology, then the corresponding algebraic cycle

Π := ∑_{j=1}^{t} β_{j} · Π_{j} ∈ CH^{d}(V_{1} × V_{2})(F) ⊗ E

satisfies condition 1 in the statement of Lemma 3.7. Condition 2 is verified by embedding F into one of its ℓ-adic completions F_{λ} and applying Fontaine’s comparison functor to (105) in which source and targets are deRham representations of G_{F_{λ}}. This shows that Π_{dR} induces an isomorphism on the deRham cohomology over F_{λ} ⊗ E, and part 2 follows. □

The following proposition (in which, to ease notations, we identify differential forms with their image in de Rham cohomology) justifies Conjecture 7 of the Introduction. Notations are as in Section 3.2 and 3.3, with ψ a Hecke character of infinity type (r + 1, 0).

Proposition 3.8. If the Tate conjecture is true for W_{r} × A^{r+1}, then there is an algebraic cycle class Π^{?} ∈ CH^{r+1}(W_{r} × A^{r+1})(F) ⊗ E_{ψ} satisfying

(107)  Π_{dR}^{?∗}(ω_{A}^{r+1}) = ω_{θ_{ψ}}.

Proof. Let M_{1,F} := (A^{r+1}, e_{r+1}, 0) be the motive over F with coefficients in K attached to the Hecke character ψ_{F} = ψ_{A}^{r+1} in Section 3.2, and let M_{2,F} := (W_{r}, e_{θ_{ψ}}, 0) be the Scholl motive over F attached to the theta-series θ_{ψ} in Section 3.3. Since the ℓ-adic realisations (M_{1,F})_{ℓ} ⊗ E_{ψ} and (M_{2,F})_{ℓ} are isomorphic, Lemma 3.7 implies the existence of a correspondence Π^{?} in CH^{r+1}(W_{r} × A^{r+1})(F) ⊗ E_{ψ} which induces an isomorphism on the de Rham realisations. This isomorphism respects the Hodge filtrations and therefore sends the class [ω_{A}^{r+1}] to a non-zero E_{ψ}-rational multiple of [ω_{θ_{ψ}}]. After suitably rescaling Π^{?}, one can therefore assume that it satisfies (107). □

Note that the ambient F -variety Z := W_{r} × A^{r+1} = W_{r} × A^{r} × A in which the correspondence Π^{?} is contained is equipped with three obvious projection maps

\xymatrix{ & Z \ar[dl]_{π_{0}} \ar[d]^{π_{1}} \ar[dr]^{π_{2}} \\ W_{r} & A^{r} & A.}

Let X_{r} be the F -variety

X_{r} = W_{r} × A^{r}.

After setting

π_{01} = π_{0} × π_{1} : Z⟶X_{r}, π_{12} = π_{1} × π_{2} : Z⟶A^{r} × A,

we recall the simple (but key!) observation already made in the Introduction that Π^{?} can be viewed as a correspondence in two different ways, via the diagrams:

\xymatrix{ & Z \ar[dl]_{π_{0}} \ar[dr]^{π_{12}} &  & and &  & Z \ar[dl]_{π_{01}} \ar[dr]^{π_{2}} \\ W_{r} &  & A^{r} × A &  & X_{r} &  & A.}

In order to maintain a notational distinction between these two ways of viewing Π^{?}, the correspondence from X_{r} to A attached to the cycle Π^{?} is denoted by Φ^{?} instead of Π^{?}. It induces a natural transformation of functors on F -algebras:

(108)  Φ^{?} : CH^{r+1}(X_{r})_{0} ⊗ E_{ψ}⟶ CH^{1}(A)_{0} ⊗ E_{ψ} = A ⊗ E_{ψ},

where A ⊗ E_{ψ} is the functor from the category of F -algebras to the category of E_{ψ}-vector spaces which to L associates A(L) ⊗ E_{ψ}. The natural transformation Φ^{?} is referred to as the modular parametrisation attached to the correspondence Φ^{?}. For any F -algebra L, we will also write

(109)  Φ_{L}^{?} : CH^{r+1}(X_{r})_{0}(L) ⊗ E_{ψ}⟶A(L) ⊗ E_{ψ}

for the associated homomorphism on L-rational points (modulo torsion).

Like the class Π^{?}, the correspondence Φ^{?} also induces a functorial F E_{ψ}-linear map on de Rham cohomology, denoted

Φ_{dR}^{?∗} : H_{dR}^{1}(A/F E_{ψ})⟶H_{dR}^{2r+1}(X_{r}/F E_{ψ}).

Recall that η_{A} ∈ H_{dR}^{1}(A/F E_{ψ}) denotes the unique class in H_{dR}^{0,1}(A/F E_{ψ}) satisfying

⟨ω_{A}, η_{A}⟩ = 1.

Proposition 3.9. The image of the class ω_{A} ∈ Ω^{1}(A/F) ⊂ H_{dR}^{1}(A/F) under Φ_{dR}^{?} is given by

Φ_{dR}^{?∗}(ω_{A}) = (ω_{θ_{ψ}} ∧ η_{A}^{r}).

Proof. After writing the cycle

Π^{?} = ∑_{j} m_{j}Z_{j}

as a E_{ψ}-linear combination of codimension (r+1) subvarieties of Z, we can define the cycle class map

cl_{Π^{?}} : H_{dR}^{2r+2}(Z/F E_{ψ})⟶F E_{ψ}

by setting

cl_{Π^{?}}(ω) = ∑_{j} m_{j} cl_{Z_{j}}(ω).

By Proposition 3.8 and the construction of Π_{dR}^{?}, we have

(110)  Π_{dR}^{?}(ω_{A}^{r+1}) = ω_{θ},

and

(111)  Π_{dR}^{?}(η_{A}^{j}ω_{A}^{r+1−j}) = 0,  for 1 ≤ j ≤ r.

By definition of Π_{dR}^{?}, equation (110) can be rewritten as

(112)  cl_{Π^{?}}(π_{0}^{∗}(α) ∧ π_{12}^{∗}(λω_{A}^{r+1})) = ⟨α, λω_{θ}⟩_{W_{r}},  for all α ∈ H_{dR}^{r+1}(W_{r}/K),

while (111) shows that

(113)  cl_{Π^{?}}(π_{0}^{∗}(α) ∧ π_{12}^{∗}(λη_{A}^{j}ω_{A}^{r+1−j})) = 0,  when 1 ≤ j ≤ r.

Equation (112) can also be rewritten as

(114)  cl_{Φ^{?}}(π_{01}^{∗}(λα ∧ ω_{A}^{r}) ∧ π_{2}^{∗}(ω_{A})) = ⟨λα ∧ ω_{A}^{r}, ω_{θ} ∧ η_{A}^{r}⟩_{X_{r}},

while equation (113) implies that, for all α ∈ H_{dR}^{r+1}(W_{r}/K) and all 1 ≤ j ≤ r,

(115)  cl_{Φ^{?}}(π_{01}^{∗}(λα ∧ η_{A}^{j}ω_{A}^{r−j}) ∧ π_{2}^{∗}(ω_{A})) = 0 = ⟨λα ∧ η_{A}^{j}ω_{A}^{r−j}, ω_{θ} ∧ η_{A}^{r}⟩_{X_{r}}.

In light of the definition of the map Φ_{dR}^{?}, equations (114) and (115) imply that

Φ_{dR}^{?}(ω_{A}) = ω_{θ} ∧ η_{A}^{r}.

The proposition follows. □

Remark 3.10. In the special case that was treated in more detail in Section 2.7, the conjectural modular parametrisation Φ^{?} admits a somewhat simpler description. This is because, in that case, we may choose

F = E_{ψ} = K.

We may then further assume (without real loss of generality, as it turns out) that the Hecke character ψ is given by

(116)  ψ = ψ_{A}^{r+1},

where ψ_{A} = ψ_{0} is the K-valued Hecke character of infinity type (1, 0) that was defined in equation (91), corresponding to an elliptic curve A/K with complex multiplication by O_{K}. Under these conditions, the modular parametrisation Φ^{?} arises from a class in CH^{r+1}(X_{r} × A)(K) ⊗ K and induces a natural transformation of functors on K-algebras:

(117)  Φ^{?} : CH^{r+1}(X_{r})_{0}⟶A.

3.5. Generalised Heegner cycles and Chow-Heegner points. Recall the notation Γ := Γ_{ε}(N) ⊂ Γ_{0}(N) in (47), and the associated modular curve C classifying (generalised) elliptic curves with Γ-level structure in the sense of Definition 2.8.

Fixing a choice t of Γ-level structure on A attached to 𝔑, the datum of (A, t) determines a point P_{A} on C(F̃) for some abelian extension F̃ of K, and a canonical embedding ι_{A} of A^{r} into the fiber in W_{r} above P_{A}. More generally, if φ : A⟶A′ is an isogeny defined over F whose kernel intersects A[𝔑] trivially (i.e., an isogeny of elliptic curves with Γ-level structure), then the pair (A′, φ(t)) determines a point P_{A′} ∈ C(F̃) and an embedding ι_{φ} : (A′)^{r}⟶W_{r} which is defined over F̃. We associate to such an isogeny φ a codimension r+1 cycle Υ_{φ} on the variety 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

(118)  Δ_{φ} := ϵ_{X}Υ_{φ} ∈ CH^{r+1}(X_{r})_{0}(F̃),

where ϵ_{X} is the idempotent given in equation (51) of [BDP], viewed as an element of the ring Corr^{0}(X_{r}, X_{r}) of algebraic correspondences from X_{r} to itself.

We will now assume that the field F has been chosen large enough so that it contains F̃ as well as the field of definition of A.

Definition 3.11. The Chow-Heegner point attached to the data (ψ, φ) is the point

P_{ψ}^{?}(φ) := Φ_{F}^{?} (Δ_{φ}) ∈ A(F) ⊗ E_{ψ}.

Note that this definition is only a conjectural one, since the existence of the homomorphism Φ_{F}^{?} depends on the existence of the algebraic cycle Π^{?}.

We now discuss some specific examples of φ that will be relevant to us. Let c be a positive integer as in Section 2.2. An isogeny φ_{0} : A⟶A_{0} is said to be a primitive isogeny of conductor c if it is of degree c and if the endomorphism ring End(A_{0}) is isomorphic to the order O_{c} in K of conductor c. The kernel of a primitive isogeny necessarily intersects A[𝔑] trivially, i.e., such a φ_{0} is an isogeny of elliptic curves with Γ-level structure. The corresponding Chow-Heegner point P_{ψ}^{?}(φ_{0}) is said to be of conductor c.

Once φ_{0} is fixed, one can also consider an infinite collection of Chow-Heegner points indexed by certain projective O_{c}-submodules of O_{c}. More precisely, let 𝔞 be such a projective module for which

A_{0}[𝔞] ∩ φ_{0}(A[𝔑]) = 0,

and let

φ_{𝔞} : A_{0}⟶A_{𝔞} := A_{0}/A_{0}[𝔞]

denote the canonical isogeny of elliptic curves with Γ-level structure given by the theory of complex multiplication. Since the isogeny φ_{𝔞} is defined over F, the Chow-Heegner point

P_{ψ}^{?}(𝔞) := P_{ψ}^{?}(φ_{𝔞}φ_{0}) = Φ_{F}^{?} (Δ_{𝔞}), where Δ_{𝔞} = Δ_{φ_{𝔞}φ_{0}},

belongs to A(F) ⊗ E_{ψ} as well.

Lemma 3.12. For all elements λ ∈ O_{c} which are prime to 𝔑, we have

P_{ψ}^{?}(λ𝔞) = ε(λ mod 𝔑)λ^{r}P_{ψ}^{?}(𝔞) in A(F) ⊗ E_{ψ}.

More generally, for any 𝔟,

φ_{𝔞}(P_{ψ}^{?}(𝔞𝔟)) = ψ(𝔞)P_{ψ}^{?}(𝔟)^{σ_{𝔞}}.

Proof. Let P_{𝔞} be the point of C(F) attached to the elliptic curve A_{𝔞} with Γ-level structure, and recall that π^{−1}(P_{𝔞}) is the fiber above P_{𝔞} for the natural projection π : X_{r}⟶C. The algebraic cycle

Δ_{λ𝔞} − ε(λ)λ^{r}Δ_{𝔞}

is entirely supported in the fiber π^{−1}(P_{𝔞}), and its image in the homology of this fiber under the cycle class map is 0. The result follows from this using the fact that the image of a cycle Δ supported on a fiber π^{−1}(P) depends only on the point P and on the image of Δ in the homology of the fiber. The proof of the general case is similar. □

Let χ be a Hecke character of K of infinity type (r, 0) such that χN_{K} belongs to Σ_{cc}^{(1)}(c, 𝔑, ε). As before, let E_{ψ,χ} denote the field generated by the values of ψ and χ. By Lemma 3.12, the expression

χ(𝔞)^{−1}P_{ψ}^{?}(𝔞) ∈ A(F) ⊗ E_{ψ,χ}

depends only on the image of 𝔞 in the class group G_{c} := Pic(O_{c}). Hence we can define the Chow-Heegner point attached to the theta-series θ_{ψ} and the character χ by summing over this class group:

(119)  P_{ψ}^{?}(χ) := ∑_{𝔞∈Pic(O_{c})} χ^{−1}(𝔞)P_{ψ}^{?}(𝔞) ∈ A(F) ⊗ E_{ψ,χ}.

The Chow-Heegner point P_{ψ}^{?}(χ) thus defined belongs (conjecturally) to A(F) ⊗ E_{ψ,χ}.

3.6. A special case. We now specialise the Chow-Heegner point construction to a simple but illustrative case, in which the hypotheses introduced in Section 2.7 are imposed. We further assume

(1) The character ψ is of the form ψ_{A}^{r+1}, as in (116), so that the modular parametrisation Φ^{?} gives a homomorphism from CH^{r+1}(X_{r})(K) to A(K) ⊗ ℚ, as in (117).

(2) The integer r is odd. This implies that ψ is an unramified Hecke character of infinity type (r+1, 0) with values in K, and that its associated theta series θ_{ψ} belongs to S_{r+2}(Γ_{0}(D), ε_{K}).

(3) The character χ is a Hecke character of infinity type (r, 0), and

χN_{K} belongs to Σ_{cc}^{(1)}(c, \sqrt{−D}, ε_{K}),

with c prime to D. A direct modification of the proof of Lemma 2.27 shows that any such χ can be written as

χ = ψ_{A}^{r}χ_{0}^{−1},

where χ_{0} is a ring class character of K of conductor dividing c.

Under these conditions, we have

Γ = Γ_{ε_{K}} (D) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(D) such that ε_{K}(a) = 1\} .

Furthermore, the action of G_{K} on the cyclic group A[\sqrt{−D}](K̄) is via the D-th cyclotomic character, and therefore a Γ-level structure of Heegner type on the curve A is necessarily defined over K. The corresponding Γ-level structures on A_{0} and on A_{𝔞} are therefore defined over the ring class field H_{c}. It follows that the generalised Heegner cycles Δ_{φ} belong to CH^{r+1}(X_{r})_{0}(H_{c}), for any isogeny φ of conductor c, and therefore–assuming the existence of Φ^{?}–that

P_{ψ}^{?}(𝔞) belongs to A(H_{c}) ⊗ ℚ, P_{ψ}^{?}(χ) belongs to (A(H_{c}) ⊗_{O_{K}} E_{χ})^{χ_{0}}.

4. Chow-Heegner points over ℂ_{p}

4.1. The p-adic Abel-Jacobi map. The construction of the point P_{ψ}^{?}(χ) is only conjectural since it depends on the existence of the cycle Π^{?} and the corresponding map Φ^{?}. In order to obtain unconditional results, we will replace the conjectural map Φ_{F}^{?} by its analogue Φ_{F}^{et} in p-adic étale cohomology mentioned in equation (22) of the Introduction, and studied in detail in Chapter 4 of [BDP].

Fix a rational prime p which does not divide the level N of θ_{ψ}. The global cohomology class

κ_{ψ}(φ) := Φ_{F}^{et}(Δ_{φ}) ∈ H^{1}(F, V_{p}(A) ⊗ E_{ψ})

belongs to the pro-p Selmer group of A over F (tensored with E_{ψ}), and is defined independently of any conjectures. Furthermore, if the correspondence Φ_{F}^{?} exists, then (23) implies that

κ_{ψ}(φ) = δ(P_{ψ}^{?}(φ)),

where

δ : A(F) ⊗ E_{ψ}⟶H^{1}(F, V_{p}(A) ⊗ E_{ψ})

is the connecting homomorphism of Kummer theory mentioned in (24).

The topological closure of F in ℂ_{p} (relative to the fixed embedding F ⟶ℂ_{p}) is a finite extension of ℚ_{p}, and shall be denoted F_{p}. Since κ_{ψ}(φ) belongs to the Selmer group of A over F, there is a local point in A(F_{p}) ⊗ E_{ψ}, denoted P_{ψ}^{(p)}(φ), such that

κ_{ψ}(φ)_{|G_{Fp}} = δ(P_{ψ}^{(p)}(φ)).

More generally, the image of Φ_{F}^{et} is contained in the Selmer group of A over F, and hence there exists a map

Φ_{F}^{(p)} : CH^{r+1}(X_{r})_{0}(F_{p})⟶A(F_{p}) ⊗ E_{ψ}

such that

Φ_{F}^{(p)}(Δ_{φ}) = P_{ψ}^{(p)}(φ).

The map Φ_{F}^{(p)} is the p-adic counterpart of the conjectural map Φ_{F}^{?} .

In light of Proposition 3.8 and of the construction of Chow-Heegner points given in Definition 3.11, the following conjecture is a concrete consequence of the Tate (or Hodge) conjecture for the variety X_{r} × A.

Conjecture 4.1. The local elements P_{ψ}^{(p)}(φ) ∈ A(F_{p}) ⊗ E_{ψ} belong to A(F) ⊗_{O_{K}} E_{ψ}.

The goal of this chapter is to exploit the connection between the local points P_{ψ}^{(p)}(φ) and special values of p-adic L-functions to supply some evidence for Conjecture 4.1.

We begin by relating P_{ψ}^{(p)}(φ) to p-adic Abel-Jacobi maps. The p-adic Abel-Jacobi map attached to the elliptic curve A/F_{p} is a homomorphism

(120)  AJ_{A} : CH^{1}(A)_{0}(F_{p})⟶Ω^{1}(A/F_{p})^{∨},

where the superscript ∨ denotes the F_{p}-linear dual. Under the identification of the Chow group CH^{1}(A)_{0}(F_{p}) with A(F_{p}), it is determined by the relation

(121)  AJ_{A}(P)(ω) = log_{ω}(P),

where ω is any regular differential on A over F_{p} and

log_{ω} : A(F_{p}) ⊗ ℚ⟶F_{p}

denotes the formal group logarithm on A attached to this choice of regular differential. It can be extended by E_{ψ}-linearity to a map from A(F_{p}) ⊗ E_{ψ} to F_{p}E_{ψ}.

There is also a p-adic Abel-Jacobi map on null-homologous algebraic cycles

AJ_{X_{r}} : CH^{r+1}(X_{r})_{0}(F_{p})⟶ Fil^{r+1} H_{dR}^{2r+1}(X_{r}/F_{p})^{∨}

attached to the variety X_{r}, where Fil^{j} refers to the j-th step in the Hodge filtration on algebraic de Rham cohomology. Details on the definition of AJ_{X_{r}} are recalled in Section 4.4 of [BDP], where it is explained how AJ_{X_{r}} can be calculated via p-adic integration.

The functoriality of the Abel-Jacobi maps under correspondences is expressed in the following commutative diagram relating AJ_{A} and AJ_{X_{r}}:

(122)  \xymatrix{CH^{r+1}(X_{r})_{0}(F_{p}) \ar[d]^{Φ_{F}^{(p)}} \ar[r]^{^{AJ^{Xr}}} & Fil^{r+1} H_{dR}^{2r+1}(X_{r}/F_{p})^{∨} \ar[d]_{(Φ_{dR}^{?∗})^{∨}} \\ A(F_{p}) ⊗ E_{ψ} \ar[r]^{_{AJ_{A}}} & Ω^{1}(A/F_{p})^{∨} ⊗ E_{ψ}.}

Proposition 4.2. For all isogenies φ : (A, t_{A}, ω_{A})⟶(A′, t′, ω′) of elliptic curves with Γ-level structure,

log_{ω_{A}}(P_{ψ}^{(p)}(φ)) = AJ_{X_{r}}(Δ_{φ})(ω_{θ} ∧ η_{A}^{r}).

Proof. By equation (121) and the definition of P_{ψ}^{(p)}(φ),

(123)  log_{ω_{A}}(P_{ψ}^{(p)}(φ)) = AJ_{A}(P_{ψ}^{(p)}(φ))(ω_{A}) = AJ_{A}(Φ_{F}^{(p)}(Δ_{φ}))(ω_{A}).

The commutative diagram (122) combined with Proposition 3.9 shows that

(124)  AJ_{A}(Φ_{F}^{(p)}Δ_{φ})(ω_{A}) = AJ_{X_{r}}(Δ_{φ})(Φ_{dR}^{?∗}ω_{A}) = AJ_{X_{r}}(Δ_{φ})(ω_{θ_{ψ}} ∧ η_{A}^{r}).

Proposition 4.2 now follows from (123) and (124). □

We will study the local points P_{ψ}^{(p)}(φ) via the formula of Proposition 4.2, whose terms do not depend on the conjectural existence of Π^{?}.

4.2. Rationality of Chow-Heegner points over ℂ_{p}. Recall that F_{ψ,χ} is the subfield of ℚ̄ generated over K by F and E_{ψ,χ}, and that ν = ψχ^{−1} is the self-dual Hecke character of K of infinity type (1, 0) attached to (ψ, χ). The main result of this section is

Theorem 4.3. There exists a global element P_{ψ}(χ) ∈ A(F) ⊗ E_{ψ,χ} such that

log_{ω_{A}}^{2}(P_{ψ}^{(p)}(χ)) = log_{ω_{A}}^{2}(P_{ψ}(χ)) (mod F_{ψ,χ}^{×}),

for all regular differentials ω_{A} ∈ Ω^{1}(A/F). This element is non-zero if and only if

L′(ν, 1) ≠ 0  and  L(ψχ^{∗−1}, 1) ≠ 0.

Proof. The proof proceeds along the same lines as (but is simpler than) the proof of Theorem 4.4 below. This proof applies to a more special setting but derives a more precise result, in which it becomes necessary to keep a more careful track of the fields of scalars involved. To prove Theorem 4.3, it is therefore enough to rewrite the proof of Theorem 4.4 with E_{χ}^{∗} replaced by F_{ψ,χ}^{×} and (ψ_{A}^{r+1}, ψ_{A}^{r}χ_{0}) replaced by (ψ, χ). Note that equations (128) and (129) hold modulo the larger group F_{ψ,χ}^{×} without the Gauss sum factors which can therefore be ignored. □

4.3. A special case. We now place ourselves in the setting of Section 3.6, in which

ψ = ψ_{A}^{r+1}, χ = ψ_{A}^{r}χ_{0}

where χ_{0} is a ring class character of K of conductor c, and we set

P_{A,r}^{(p)}(χ_{0}) := P_{ψ_{A}^{r+1}}^{(p)}(ψ_{A}^{r}χ_{0}) = P_{ψ}^{(p)}(χ).

Recall the definition of the χ_{0}-component (A(H_{c}) ⊗ E_{χ})^{χ_{0}} of the Mordell-Weil group over the ring class field H_{c} that was given in (92).

Theorem 4.4. There exists a global point P_{A,r}(χ_{0}) ∈ (A(H_{c}) ⊗ E_{χ})^{χ_{0}} satisfying

log_{ω_{A}}^{2}(P_{A,r}^{(p)}(χ_{0})) = log_{ω_{A}}^{2}(P_{A,r}(χ_{0})) (mod E_{χ}^{×}).

Furthermore, the point P_{A,r}(χ_{0}) is of infinite order if and only if

L′(ψ_{A}χ_{0}^{−1}, 1) ≠ 0, L(ψ_{A}^{2r+1}χ_{0}, r+1) ≠ 0.

Proof. By Proposition 4.2,

(125)  log_{ω_{A}}(P_{A,r}^{(p)}(χ)) = AJ_{X_{r}}(Δ_{ψ}(χ))(ω_{θ_{ψ}} ∧ η_{A}^{r}).

Theorem 6.13 of [BDP] with f = θ_{ψ} and j = 0 gives

(126)  AJ_{X_{r}}(Δ_{ψ}(χ))(ω_{θ_{ψ}} ∧ η_{A}^{r})^{2} = \frac{L_{p}(θ_{ψ}, χN_{K})}{Ω_{p}(A)^{2r}} (mod E_{χ}^{×}).

The fact that θ_{ψ} has Fourier coefficients in K and that its Nebentype character ε_{K} is trivial when restricted to K implies that the field E_{ψ,χ,ε_{K}} occurring in Corollary 2.14 is equal to E_{χ}. Therefore, this corollary implies that

\frac{L_{p}(θ_{ψ}, χN_{K})}{Ω_{p}(A)^{2r}} = ℒ_{p}(ψ^{−1}χN_{K}) × \frac{ℒ_{p}(ψ^{∗−1}χN_{K})}{Ω_{p}(A)^{2r}} (mod E_{χ}^{×})
(127)  = \frac{ℒ_{p}(ν^{∗})}{Ω_{p}(A)^{−1}} × \frac{ℒ_{p}(ψ_{A}^{2r+1}χ_{0}N_{K}^{−r})}{Ω_{p}(A)^{2r+1}} (mod E_{χ}^{×}).

The character ν^{∗} = ψ_{A}^{∗}χ_{0} lies in Σ_{sd}^{(1)}(c) and is of type (0, 1). Hence, Theorem 2.28 can be invoked. This theorem gives a global element P_{A}(χ_{0}) ∈ (A(H_{c}) ⊗ E_{χ})^{χ_{0}} which is of infinite order if and only if L′(ψ_{A}χ_{0}^{−1}, 1) ≠ 0, and satisfies

(128)  ℒ_{p}(ψ_{A}^{∗}χ_{0}) = Ω_{p}(A)^{−1}𝔤(χ_{0}) log_{ω_{A}}^{2}(P_{A}(χ_{0})) (mod E_{χ}^{×}).

Furthermore, the character ψ_{A}^{2r+1}χ_{0}N_{K}^{−r} belongs to the domain Σ_{sd}^{(2)}(c) of classical interpolation for the Katz p-adic L-function. Proposition 1.16 and Lemma 1.15 show that the p-adic period attached to this central critical character is given by

(129)  Ω_{p}((ψ_{A}^{2r+1}χ_{0}N_{K}^{−r})^{∗}) = Ω_{p}(A)^{2r+1}𝔤(χ_{0})^{−1} (mod E_{χ}^{∗}).

Therefore, Theorem 2.4 implies that, up to multiplication by a non-zero element of E_{χ},

(130)  ℒ_{p}(ψ_{A}^{2r+1}χ_{0}N_{K}^{−r}) = \{ \begin{matrix}0 & if L(ψ_{A}^{2r+1}χ_{0}, r+1) = 0, \\ Ω_{p}(A)^{2r+1}𝔤(χ_{0})^{−1} & otherwise. \end{matrix}

After setting

(131)  P_{A,r}(χ_{0}) = \{ \begin{matrix}0 & if L(ψ_{A}^{2r+1}χ_{0}, r+1) = 0, \\ P_{A}(χ_{0}) & otherwise, \end{matrix}

equations (128) and (130) can be used to rewrite (127) as

(132)  \frac{L_{p}(θ_{ψ}, χN_{K})}{Ω_{p}(A)^{2r}} = log_{ω_{A}}^{2}(P_{A,r}(χ_{0})) (mod E_{χ}^{×}).

Theorem 4.4 now follows by combining (125), (126) and (132). □

We now specialise the setting even further by assuming that χ_{0} = 1 is the trivial character, so that ψ = ψ_{A}^{r+1} and χ = ψ_{A}^{r}, and set

P_{A,r}^{(p)} := P_{ψ_{A}^{r+1}}^{(p)}(ψ_{A}^{r})).

In this case, the coefficient field E_{χ} is equal to K, and Theorem 4.4 asserts the existence of a point P_{A,r} ∈ A(K) such that

log_{ω_{A}}^{2}(P_{A,r}^{(p)}) = log_{ω_{A}}^{2}(P_{A,r}) (mod K^{×}).

It is instructive to refine the argument used in the proof of Theorem 4.4 to resolve the ambiguity by the non-zero scalar in K^{×}, in order to examine the dependence on r of the local point P_{A,r}^{(p)}. This is the content of the next result.

Theorem 4.5. For all odd r ≥ 1, the Chow-Heegner point P_{A,r}^{(p)} belongs to A(K) ⊗ K and is given by the formula

(133)  log_{ω_{A}}^{2}(P_{A,r}^{(p)}) = ℓ(r) · log_{ω_{A}}^{2}(P_{A}),

CHOW-HEEGNER POINTS ON CM ELLIPTIC CURVES  AND VALUES OF p-ADIC L-FUNCTIONS  41

where ℓ(r) ∈ ℤ satisfies

ℓ(r) = ±\frac{r!(2π)^{r}}{(2\sqrt{D})^{r}Ω(A)^{2r+1}} L(ψ_{A}^{2r+1}, r + 1),

and P_{A} is a generator of A(K) ⊗ K depending only on A but not on r. Proof. As in the proof of Theorem 4.4, we combine (125) and Theorem 6.13 of [BDP] with (f, j) = (θ_{ψ_{A}^{r+1}}, 0) and χN_{K} = ψ_{A}^{r}N_{K} playing the role of χ, to obtain

(134)  log_{ω_{A}}^{2}(P_{ψ}(χ)) = (1 − (pχ(𝔭̄))^{−1}a_{p}(θ_{ψ}) + (pχ(𝔭̄))^{−2}p^{r+1})^{−2} \frac{L_{p}(θ_{ψ}, χN_{K})}{Ω_{p}(A)^{2r}}.

Since χ(𝔭̄) = ψ_{A}(𝔭̄)^{r} and a_{p}(θ_{ψ}) = ψ_{A}^{r+1}(𝔭̄) + ψ_{A}^{r+1}(𝔭), the Euler factor appearing in (134) is given by

(1 − ψ_{A}^{−1}(𝔭))^{−2}(1 − ψ_{A}^{2r+1}(𝔭)p^{−r−1})^{−2}.

Therefore,

(135)  log_{ω_{A}}^{2}(P_{ψ}(χ)) = (1 − ψ_{A}^{−1}(𝔭))^{−2}(1 − ψ_{A}^{2r+1}(𝔭)p^{−r−1})^{−2} \frac{L_{p}(θ_{ψ}, χN_{K})}{Ω_{p}(A)^{2r}}.

On the other hand, by Theorem 2.13 with c = 1 and j = 0

(136)  \frac{L_{p}(θ_{ψ}, χN_{K})}{Ω_{p}(A)^{2r}} = \frac{w(θ_{ψ}, χ)^{−1}}{2^{r}} × ℒ_{p}(ψ_{A}^{∗}) × \frac{ℒ_{p}(ψ_{A}^{2r+1}N_{K}^{−r})}{Ω_{p}(A)^{2r}} .

By Lemma 6.3 of [BDP], the norm 1 scalar w(θ_{ψ}, χ) belongs to K, and is only divisible by the primes above \sqrt{−D}. Therefore it is a unit in O_{K}, and hence is equal to ±1. Therefore

(137)  \frac{L_{p}(θ_{ψ}, χN_{K})}{Ω_{p}(A)^{2r}} = \frac{±1}{2^{r}} × \frac{ℒ_{p}(ψ_{A}^{∗})}{Ω_{p}(A)^{−1}} × \frac{ℒ_{p}(ψ_{A}^{2r+1}N_{K}^{−r})}{Ω_{p}(A)^{2r+1}} .

Let P_{A} = P_{A}(1) ∈ A(K) ⊗ K be as in (128), but chosen specifically so that

(138)  \frac{ℒ_{p}(ψ_{A}^{∗})}{Ω_{p}(A)^{−1}} = (1 − ψ_{A}^{−1}(𝔭))^{2} log_{ω_{A}}^{2}(P_{A}).

By the interpolation property for the Katz L-function given in Proposition 2.3 with j = r and ν = ψ_{A}^{2r+1}N_{K}^{−r} = ψ_{A}^{r+1}ψ_{A}^{∗−r},

(139)  \frac{ℒ_{p}(ψ_{A}^{2r+1}N_{K}^{−r})}{Ω_{p}(A)^{2r+1}} = (1 − ψ_{A}(𝔭)^{2r+1}p^{−r−1})^{2} × \frac{r!(2π)^{r}L((ψ_{A}^{∗})^{2r+1}N_{K}^{−r−1}, 0)}{\sqrt{D}^{r}Ω(A)^{2r+1}}.

After substituting equations (138) and (139) into (137), and using the fact that

L((ψ_{A}^{∗})^{2r+1}N_{K}^{−r−1}, 0) = L(ψ_{A}^{2r+1}, r+1),

we find

(1 − ψ_{A}^{−1}(𝔭))^{−2}(1 − ψ_{A}(𝔭)^{2r+1}p^{−r−1})^{−2} × \frac{L_{p}(θ_{ψ}, χN_{K})}{Ω_{p}(A)^{2r}}
= \frac{±1}{2^{r}} log_{ω_{A}}^{2}(P_{A}) × \frac{r!(2π)^{r}L(ψ_{A}^{2r+1}, r+1)}{\sqrt{D}^{r}Ω(A)^{2r+1}}.

Hence, by (135), we obtain

log_{ω_{A}}^{2}(P_{ψ}(χ)) = ±\frac{r!(2π)^{r}}{(2\sqrt{D})^{r}Ω(A)^{2r+1}} × L(ψ_{A}^{2r+1}, r + 1) × log_{ω_{A}}^{2}(P_{A})

The result follows. □

5. Chow-Heegner points over ℂ

5.1. The complex Abel-Jacobi map. For simplicity, we will confine ourselves in this section to working under the hypotheses that were made in Section 2.7 where K is assumed in particular to have discriminant −D, with

D ∈ S := {7, 11, 19, 43, 67, 163}.

The difficulty in computing the modular parametrisation Φ^{?} and the resulting Chow-Heegner points arises from the fact that it is hard in general to explicitly produce the correspondence Φ^{?}, or even to ascertain its existence. However, when F = ℂ, the Chow-Heegner points P_{ψ}^{?}(φ) can be evaluated numerically in practice via integration of smooth differential forms on X_{r}(ℂ). More precisely, let

(140)  AJ_{A}^{∞} : CH^{1}(A)_{0}(ℂ)⟶\frac{Fil^{1} H_{dR}^{1}(A/ℂ)^{∨}}{Im H_{1}(A(ℂ), ℤ)}

be the classical complex Abel-Jacobi map attached to A, where the superscript ∨ now denotes the complex linear dual. The map AJ_{A}^{∞} is defined by the rule

(141)  AJ_{A}^{∞}(Δ)(ω) = ∫_{∂^{−1}Δ} ω,

the integral on the right being taken over any one-chain on A(ℂ) having the degree zero divisor Δ as boundary. This classical Abel-Jacobi map admits a higher dimensional generalisation for null-homologous cycles on X_{r} introduced by Griffiths and Weil:

(142)  AJ_{X_{r}}^{∞} : CH^{r+1}(X_{r})_{0}(ℂ)⟶\frac{Fil^{r+1} H_{dR}^{2r+1}(X_{r}/ℂ)^{∨}}{Im H_{2r+1}(X_{r}(ℂ), ℤ)} ,

defined just as in (141), but where AJ_{X_{r}}^{∞}(Δ)(ω) is now defined by integrating any smooth representative of the de Rham cohomology class ω against a (2r + 1)-chain on X_{r}(ℂ) having Δ as boundary. (Cf. the description in Chapter 3 of [BDP] for example.) The map AJ_{X_{r}}^{∞} is the complex analogue of the p-adic Abel-Jacobi map AJ_{Xr} that was introduced and studied in Section 4.

The functoriality of the Abel-Jacobi maps under correspondences is expressed in the following commutative diagram which is the complex counterpart of (122):

(143)  [figure: CH^{r+1}(X_{r})_{0}(ℂ); Φ_{ℂ}^{?}; ^{AJ{∞ \atop Xr}}; \frac{Fil^{r+1} H_{dR}^{2r+1}(X_{r}/ℂ)^{∨}}{Im H_{2r+1}(X_{r}(ℂ),ℤ)}; (Φ_{dR}^{?∗})^{∨}; CH^{1}(A)_{0}(ℂ); _{AJ{∞ \atop A}}; \frac{Ω^{1}(A/ℂ)^{∨}}{Im H_{1}(A(ℂ),ℤ)}.]

Since AJ_{A}^{∞} is an isomorphism, we can simply define the complex analogue Φ_{ℂ} of Φ_{F}^{(p)} as the unique map from CH^{r+1}(X_{r})_{0}(ℂ) to A(ℂ) for which the diagram above (with Φ_{ℂ}^{?} replaced by Φ_{ℂ}) commutes.

Recall the distinguished element ω_{A} of Ω^{1}(A/ℂ) and let

Λ_{A} := {∫_{γ} ω_{A}, γ ∈ H_{1}(A(ℂ), ℤ)} ⊂ ℂ

be the associated period lattice. Recall that φ : (A, t_{A}, ω_{A})⟶(A′, t′, ω′) is an isogeny of elliptic curves with Γ-level structure if

φ(t_{A}) = t′ and φ^{∗}(ω′) = ω_{A}.

The following proposition, which is the complex counterpart of Proposition 4.2, expresses the Abel-Jacobi image of P_{ψ}^{?}(φ) in terms of the Griffiths higher Abel-Jacobi map.

Proposition 5.1. For all isogenies φ : (A, t_{A}, ω_{A})⟶(A′, t′, ω′) of elliptic curves with Γ-level structure,

AJ_{A}^{∞}(P_{ψ}^{?}(φ))(ω_{A}) = AJ_{X_{r}}^{∞}(Δ_{φ})(ω_{θ_{ψ}} ∧ η_{A}^{r}) (mod Λ_{ω_{A}}).

Proof. The proof is the same as for Proposition 4.2: By definition of P_{ψ}(φ) combined with the commutative diagram (143),

AJ_{A}^{∞}(P_{ψ}(φ))(ω_{A}) = AJ_{A}^{∞}(Φ_{ℂ}^{?}Δ_{φ})(ω_{A}) = AJ_{X_{r}}^{∞}(Δ_{φ})(Φ_{dR}^{?∗}ω_{A}).

On the other hand, by Proposition 3.9,

Φ_{dR}^{?∗}ω_{A} = ω_{θ_{ψ}} ∧ η_{A}^{r}.

Proposition 5.1 follows. □

We now turn to giving an explicit formula for the right hand side of the equation in Proposition 5.1. To do this, let Λ_{ω′} ⊂ ℂ be the period lattice associated to the regular differential ω′ on A′. Note that Λ_{ω_{A}} is contained in Λ_{ω′} with index deg(φ).

Definition 5.2. A basis (ω_{1}, ω_{2}) of Λ_{ω′} is said to be admissible relative to (A′, t′) if (1) The ratio τ := ω_{1}/ω_{2} has positive imaginary part;

(2) via the identification \frac{1}{N} Λ_{ω′}/Λ_{ω′} = A′(ℂ)[N], the N-torsion point ω_{2}/N belongs to the orbit t′.

Given an arbitrary cusp form f ∈ S_{r+2}(Γ_{0}(N), ε), consider the cohomology class

ω_{f} ∧ η_{A}^{r} = (2πi)^{r+1}f(z)dzdw^{r} ∧ η_{A}^{r} ∈ Fil^{r+1} H_{dR}^{2r+1}(X_{r}/ℂ).

Proposition 5.3. Let Δ_{φ} be the generalised Heegner cycle corresponding to the isogeny

φ : (A, t_{A}, ω_{A})⟶(A′, t′, ω′)

of elliptic curves with Γ-level structure, let (ω_{1}, ω_{2}) be an admissible basis for Λ_{ω′}, and let τ = ω_{1}/ω_{2}. Then

(144)  AJ_{X_{r}}^{∞}(Δ_{φ})(ω_{f} ∧ η_{A}^{r}) = ω_{2}^{−r} \frac{(2πi)^{r+1}}{(τ − τ̄)^{r}} ∫_{i∞}^{τ}(z − τ̄)^{r}f(z)dz.

Proof. We begin by observing that replacing ω_{A} by a scalar multiple λω_{A} multiplies both the left and right hand sides of (144) by λ^{−r}. Hence we may assume, after possibly rescaling Λ_{ω′}, that the admissible basis (ω_{1}, ω_{2}) is of the form (2πiτ, 2πi) with τ ∈ H. The case j = 0 in Theorem 3.15 of [BDP] then implies that

AJ_{X_{r}}^{∞}(Δ_{φ})(ω_{f} ∧ η_{A}^{r}) = \frac{2πi}{(τ − τ̄)^{r}} ∫_{i∞}^{τ}(z − τ̄)^{r}f(z)dz
= ω_{2}^{−r} \frac{(2πi)^{r+1}}{(τ − τ̄)^{r}} ∫_{i∞}^{τ}(z − τ̄)^{r}f(z)dz.

The Proposition follows. □

Theorem 5.4. Let P_{ψ}(φ) be the Chow-Heegner point corresponding to the generalised Heegner cycle Δ_{φ}. With notations as in Proposition 5.3,

(145)  AJ_{A}^{∞}(P_{ψ}(φ))(ω_{A}) = ω_{2}^{−r} \frac{(2πi)^{r+1}}{(τ − τ̄)^{r}} ∫_{i∞}^{τ}(z − τ̄)^{r}θ_{ψ}(z)dz (mod Λ_{ω_{A}}).

Proof. This is an immediate corollary of Propositions 5.1 and 5.3. □

5.2. Numerical experiments. We now describe some numerical evaluations of Chow-Heegner points. As it stands, the elliptic curve A of conductor D^{2} attached to the canonical Hecke character ψ_{A} = ψ_{0} is only determined up to isogeny, and we pin it down by specifying that A is described by the minimal Weierstrass equation

A : y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6},

where the coefficients a_{1}, …, a_{6} are given in Table 1 below.

\begin{matrix}D & a_{1} & a_{2} & a_{3} & a_{4} & a_{6} & Ω(A) & P_{A} \\ 7 & 1 & −1 & 0 & −107 & 552 & 1.93331170 … & − \\ 11 & 0 & −1 & 1 & −7 & 10 & 4.80242132 … & (4, 5) \\ 19 & 0 & 0 & 1 & −38 & 90 & 4.19055001 … & (0, 9) \\ 43 & 0 & 0 & 1 & −860 & 9707 & 2.89054107 … & (17, 0) \\ 67 & 0 & 0 & 1 & −7370 & 243528 & 2.10882279 … & (\frac{201}{4}, \frac{−71}{8}) \\ 163 & 0 & 0 & 1 & −2174420 & 1234136692 & 0.79364722 … & (850, −69) \end{matrix}

Table 1: The canonical elliptic curve A

The penultimate column in Table 1 gives an approximate value for the positive real period Ω(A) attached to the elliptic curve A and its Néron differential ω_{A}. In all cases, the Néron lattice Λ_{A} attached to (A, ω_{A}) is generated by the periods

(146)  ω_{1} := (\frac{D + \sqrt{−D}}{2D} ) Ω(A),  ω_{2} := Ω(A),

and (ω_{1}, ω_{2}) is an admissible basis for Λ_{A} in the sense of Definition 5.2.

The elliptic curve A has Mordell-Weil rank 0 over ℚ when D = 7 and rank one otherwise. A specific generator P_{A} for A(ℚ) ⊗ ℚ is given in the last column of Table 1.

Recall that ψ_{A} is the Hecke character attached to A, satisfying

L(ψ_{A}, s) = L(A, s).

As in Theorem 4.5, we let ψ = ψ_{A}^{r+1} with r ≥ 1 an odd integer, so that the associated theta series θ_{ψ} belongs to S_{r+2}(Γ_{0}(D), ε_{K}) and therefore satisfies the Heegner hypothesis.

After letting Γ = Γ_{ε_{K}} (D), we observe as before that the elliptic curve A has two canonical Γ-level structures defined over K, corresponding to the two orbits of ((ℤ/Dℤ)^{×})^{2} acting on A[\sqrt{−D}]^{∗}. Fix such a Γ-level structure t_{A} once and for all. Let C := X̃_{0}(D) be the coarse moduli space classifying elliptic curves with Γ-level structure. The forgetful functor which to (E, t_{E}) associates (E, ⟨t_{E}⟩), where ⟨t_{E}⟩ denotes the cyclic subgroup of order D generated by t_{E}, induces an isomorphism from X̃_{0}(D) to X_{0}(D). But the moduli description for X̃_{0}(D) is somewhat finer, since an elliptic curve E with Γ-level structure has no non-trivial automorphisms, at least when End(E) ≠ ℤ[i] or ℤ[\frac{1+\sqrt{−3}}{2}]. (In particular, X̃_{0}(D) is a fine moduli space when D ≡ 11 (mod 12).) Denote by W_{r} the r-th Kuga-Sato variety over the modular curve X̃_{0}(D) and let X_{r} = W_{r} × A^{r}.

Assume that an algebraic correspondence Π^{?} from W_{r} to A^{r+1} as in Proposition 3.8 exists. Assume further that it is defined over ℚ, and has integral coefficients, i.e., that it belongs to Corr^{0}(W_{r}, A^{r+1}) rather than just Corr^{0}(W_{r}, A^{r+1}) ⊗ ℚ. In that case it gives rise to a parametrisation

Φ^{?} : CH^{r+1}(X_{r})_{0}⟶A,

where X_{r} = W_{r} × A^{r}.

Since the space of theta-series attached to the elliptic curve A has dimension h(−D) = 1, and since the classes of ω_{θ_{ψ}} and ω_{A} are both defined over ℚ, there is a non-zero rational scalar c satisfying

Π_{dR}^{?∗}(ω_{A}^{r+1}) = c · ω_{θ_{ψ}}.

This scalar can be viewed as playing the role of the Manin-constant in the context of the modular parametrisation of A by CH^{r+1}(X_{r})_{0}.

Question 5.5. When is it possible to choose the integral cycle Π^{?} so that c = 1?

By allowing Π^{?} to belong to Corr^{0}(W_{r}, A^{r+1}) ⊗ ℚ, we can assume, after suitably rescaling Π^{?}, that c = 1. We will assume for the rest of this chapter that Π^{?} has been rescaled in this way.

5.2.1. Chow-Heegner points of level 1. For D ∈ S := {11, 19, 43, 67, 163}, the elliptic curve A has rank 1 over ℚ. Let r ≥ 1 be an odd integer. By Theorem 5.4, the Chow-Heegner point P_{A,r} attached to the class of the diagonal Δ ⊂ (A × A)^{r} is given by

(147)  AJ_{A}^{∞}(P_{A,r})(ω_{A}) = J_{r} := ω_{2}^{−r} \frac{(2πi)^{r+1}}{(τ − τ̄)^{r}} ∫_{i∞}^{τ}(z − τ̄)^{r}θ_{ψ}(z)dz,

where (ω_{1}, ω_{2}) is the admissible basis of Λ_{A} given in (146), and τ = \frac{ω_{1}}{ω_{2}} = \frac{D+\sqrt{−D}}{2D}. Hence the complex point P_{A,r} can be computed as the natural image of the complex number J_{r} under the Weierstrass uniformisation.

We have calculated the complex points P_{A,r} for all D ∈ S and all r ≤ 15, to roughly 200 digits of decimal accuracy. The calculations indicate that

(148)  P_{A,r} =^{?} \sqrt{−D} · m_{r} · P_{A} (mod A(ℂ)[ι_{r}]),

where P_{A} is the generator of A(ℚ) ⊗ ℚ given in Table 1, and m_{r} is the rational integer listed in Table 2 below, in which the columns correspond to D ∈ S and the rows to the odd r between 1 and 15.

11  19  43  67  163
1  1  1  1  1  1
3  2  6  36  114  2172
5  −8  −16  440  6920  3513800
7  14  −186  −19026  −156282  3347376774
9  304  4176  −8352  −34999056  −238857662304
11 −352 −33984  33708960  3991188960  −3941159174330400
13 76648  545064 −2074549656  46813903656  1904546981028802344
15 274736 40959504 47714214240 −90863536574160 8287437850155973464480

Table 2: The constants m_{r} for 1 ≤ r ≤ 15.

The first 6 lines in this table, corresponding to 1 ≤ r ≤ 11, are in perfect agreement with the values that appear in the third table of Section 3.1 of [RV]. This coincidence, combined with Theorem 3.1. of [RV], suggests the following conjecture which is consistent with the p-adic formulae obtained in Theorem 4.5.

Conjecture 5.6. For all D ∈ S and all odd r ≥ 1, the Chow-Heegner point P_{A,r} belongs to A(K) ⊗ ℚ and is given by the formula

(149)  P_{A,r} = \sqrt{−D} · m_{r} · P_{A},

where m_{r} ∈ ℤ satisfies the formula

m_{r}^{2} = \frac{2r!(2π\sqrt{D})^{r}}{Ω(A)^{2r+1}} L(ψ_{A}^{2r+1}, r + 1),

and P_{A} is the generator of A(ℚ) ⊗ ℚ given in Table 1.

The fact that the experimentally observed equality in (148) only holds modulo a subgroup of A(ℂ)_{tors} may reflect the fact that the correspondence Π^{?} from W_{r} to A^{r+1} needs to be taken in Cor(W_{r}, A^{r+1}) ⊗ ℚ in general. The optimal values of ι_{r} that were observed experimentally are recorded in Table 3 below, for 1 ≤ r ≤ 31.

\begin{matrix}r & 11 & 19 & 43 & 67 & 163 & r & 11 & 19 & 43 & 67 & 163 \\ 1 & 3 & 1 & 1 & 1 & 1 & 17 & 3^{3} & 7 & 1 & 19 & 1 \\ 3 & 3 · 5 & 5 & 1 & 1 & 1 & 19 & 3 · 5^{2} & 5^{2} · 11 & 11 & 1 & 1 \\ 5 & 2 · 3^{2} & 2 · 7 & 2 & 2 & 2 & 21 & 3 · 23 & 23 & 23 & 23 & 1 \\ 7 & 2 · 7 & 5 & 1 & 1 & 1 & 23 & 3^{2} · 5 & 5 · 7 & 13 & 1 & 1 \\ 9 & 3 & 11 & 11 & 1 & 1 & 25 & 3 & 1 & 1 & 1 & 1 \\ 11 & 3^{2} · 5 & 5 · 7 & 13 & 1 & 1 & 27 & 3 · 5 & 5 & 1 & 29 & 1 \\ 13 & 3 & 1 & 1 & 1 & 1 & 29 & 3^{2} · 31 & 7 · 11 & 11 · 31 & 1 & 1 \\ 15 & 3 · 5 & 5 · 17 & 17 & 17 & 1 & 31 & 3 · 5 & 5 · 17 & 17 & 1 & 1 \end{matrix}

Table 3: The ambiguity factor ι_{r} for 1 ≤ r ≤ 31.

Remark 5.7. The data in Table 3 suggests that the term ι_{r} in (148) is only divisible by primes that are less than or equal to r + 2. One might therefore venture to guess that the primes ℓ dividing ι_{D,r} are only those for which the mod ℓ Galois representation attached to ψ_{A}^{r+1} has very small image, or perhaps non-trivial G_{K}-invariants.

5.2.2. Chow-Heegner points of prime level. We may also consider (for a fixed D and a fixed odd integer r) the Chow-Heegner points on A attached to non-trivial isogenies φ. For instance, let ℓ ≠ D be a prime. There are ℓ + 1 distinct isogenies φ_{j} : A⟶A′_{j} of degree ℓ (with j = 0, 1, …, ℓ − 1, ∞) attached to the lattices Λ′_{0}, …, Λ′_{ℓ−1}, Λ′_{∞} containing Λ_{A} with index ℓ. These lattices are generated by the admissible bases

Λ′_{j} = ℤ (\frac{ω_{1} + jω_{2}}{ℓ} ) ⊕ ℤω_{2}, Λ′_{∞} = ℤω_{1} ⊕ ℤ\frac{ω_{2}}{ℓ} .

The elliptic curves A′_{j} and the isogenies φ are defined over the ring class field H_{ℓ} of K of conductor ℓ. Let

J_{r}(ℓ, j) := ℓ^{r}ω_{2}^{−r} \frac{(2πi)^{r+1}}{(τ − τ̄)^{r}} ∫i∞^{\frac{τ+j}{ℓ}} (z − \frac{τ̄ + j}{ℓ} )^{r} θ_{D,r}(z)dz, 0 ≤ j ≤ ℓ − 1,
J_{r}(ℓ, ∞) := ε_{K}(ℓ)ω_{2}^{−r} \frac{(2πi)^{r+1}}{(τ − τ̄)^{r}} ∫_{i∞}^{ℓτ} (z − ℓτ̄)^{r}θ_{ψ}(z)dz

be the associated complex invariants and let P_{A,r}(ℓ, j) and P_{A,r}(ℓ, ∞) denote the corresponding points in ℂ/Λ_{A} = A(ℂ).

We have attempted to verify the following conjecture numerically.

Conjecture 5.8. For all ℓ ≠ D and all j ∈ P_{1}(𝔽_{ℓ}), the complex points P_{A,r}(ℓ, j) belong to the Mordell-Weil group A(H_{ℓ}) ⊗ ℚ. More precisely,

(1) If ℓ is inert in K, then Gal(H_{ℓ}/K) acts transitively on the set

{P_{A,r}(ℓ, j), j ∈ P_{1}(𝔽_{ℓ})}

of Chow-Heegner points of level ℓ.

(2) If ℓ = λλ̄ is split in K, then there exist j_{1}, j_{2} ∈ P_{1}(𝔽_{ℓ}) for which

P_{A,r}(ℓ, j_{1}) = ε_{K}(λ)λ^{r}P_{A,r}, P_{A,r}(ℓ, j_{2}) = ε_{K}(λ̄)λ̄^{r}P_{A,r},

and Gal(H_{ℓ}/K) acts transitively on the remaining set

{P_{A,r}(ℓ, j), j ∈ P_{1}(𝔽_{ℓ}) − {j_{1}, j_{2}}}

of Chow-Heegner points of level ℓ.

We have tested this prediction numerically for r = 1 and all

D ∈ S, ℓ = 2, 3, 5, 7, 11,

as well as in a few cases where r = 3. In all cases the points P_{A,r}(ℓ, j) were identified with algebraic points in A(H_{ℓ}) ⊗ ℚ, with a convincing amount of numerical accuracy. Such calculations sometimes required several hundred digits of numerical precision, together with a bit of trial and error. The necessity for this arose because Conjecture 5.8 only predicts that the points P_{A,r}(ℓ, j) belong to A(H_{ℓ}) ⊗ ℚ and not A(H_{ℓ}). So in practice, these complex points need to be multiplied by a (typically small) integer in order to belong to A(H_{ℓ}). Furthermore, the resulting global points appear (as suggested by (149) in the case ℓ = 1) to be divisible by \sqrt{−D}, and this causes their heights to be rather large. It is therefore better in practice to divide the P_{A,r}(ℓ, j) by \sqrt{−D}, which introduces a further ambiguity of A(ℂ)[\sqrt{−D}] in the resulting global point. The conjecture that was eventually tested numerically is the following non-trivial strengthening of Conjecture 5.8:

Conjecture 5.9. Given integers n ∈ ℤ^{≥1} and 0 ≤ s ≤ D − 1, let

J′_{r}(ℓ, j) = n · \frac{J_{r}(ℓ, j) − sω_{1}}{\sqrt{−D}} ,  0 ≤ j ≤ ℓ − 1,
J′_{r}(ℓ, ∞) = n · \frac{J_{r}(ℓ, ∞) − sε_{K}(ℓ)ℓ^{r}ω_{1}}{\sqrt{−D}} ,

and let P′_{A,r}(ℓ, j) ∈ A(ℂ) be the associated complex points. Then there exist n = n_{D,r} and s = s_{D,r}, depending on D and r but not on ℓ and j, for which the points P′_{r}(ℓ, j) belong to A(H_{ℓ}) and satisfy all the conclusions of Conjecture 5.8 with P_{A,r}(ℓ, j) replaced by P′_{A,r}(ℓ, j).

We now describe a few sample calculations that lend support to Conjecture 5.9.

1. The case D = 7. Consistent with the fact that the elliptic curve A has rank 0 over ℚ (and hence over K as well), the point P_{A,r} appears to be a torsion point in A(ℂ), for all 1 ≤ r ≤ 31. For example, the invariant J_{1} agrees with (ω_{1} + ω_{2})/8 to the 200 decimal digits of accuracy that were calculated. When ℓ = 2, it also appears that the quantities J_{1}(2, j) belong to \frac{1}{8}Λ_{7}. There is no reason, however, to expect the Chow-Heegner points P_{A,r}(ℓ, j) to be torsion for larger values of ℓ. Experiments suggest that the constants in Conjecture 5.9 are

n_{7,1} = 4, s_{7,1} = 0.

For example, when ℓ = 3, the ring class field of conductor ℓ is a cyclic quartic extension of K containing K(\sqrt{21}) as its quadratic subfield. In that case, the points P′_{A,1}(3, j) satisfy

P′_{A,1}(3, 0) = P′_{A,1}(3, 1) = −P′_{A,1}(3, 2) = −P′_{A,1}(3, ∞),

and agree to 600 digits of accuracy with a global point in A(ℚ(\sqrt{21})) of relatively small height, with x-coordinate given by

x = \frac{259475911175100926920835360582209388259}{41395589491845015952295204909998656004}.

2. The case D = 19. To compute the Chow-Heegner points of conductor 3 in the case D = 19 and r = 1, it appears that one can take

n_{19,1} = 1, s_{19,1} = 1.

Perhaps because of the small value of n_{19,1}, the points P′_{A,1}(ℓ, j) appear to be of relatively small height and can easily be recognized as global points, even for moderately large values of ℓ. For instance, the points P′_{A,1}(3, j) seem to have x-coordinates of the form

x = \frac{−19 ± 3\sqrt{57}}{2},

and their y-coordinates satisfying the degree 4 polynomial

x^{4} + 2x^{3} + 8124x^{2} + 8123x − 217886

whose splitting field is the ring class field H_{3} of K of conductor 3.

When ℓ = 7, which is split in K/ℚ, the ring class field H_{7} is a cyclic extension of K of degree 6. It appears that the points P′_{A,1}(7, 3) and P′_{A,1}(7, 5) belong to A(K) and are given by

P′_{A,1}(7, 3) = \frac{3 + \sqrt{−19}}{2}P_{A}, P′_{A,1}(7, 5) = \frac{3 − \sqrt{−19}}{2}P_{A}.

The 6 remaining points are grouped into three pairs of equal points,

P′_{A,1}(7, 0) = P′_{A,1}(7, 2),  P′_{A,1}(7, 1) = P′_{A,1}(7, 6),  P′_{A,1}(7, 4) = P′_{A,1}(7, ∞),

whose x and y coordinates appear to satisfy the cubic polynomials

9x^{3} + 95x^{2} + 19x − 1444, 27x^{3} − 235x^{2} + 557x + 1198

respectively. The splitting field of both of these polynomials turns out to be the cubic subfield L of the ring class field of K of conductor 7. One obtains as a by-product of this calculation 3 independent points in A(L) which are linearly independent over O_{K}. We expect that these three points give a K-basis for A(L) ⊗ ℚ (and therefore that A(L) has rank 6) but have not checked this numerically.

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