p-ADIC L-FUNCTIONS AND THE CONIVEAU FILTRATION ON CHOW GROUPS

MASSIMO BERTOLINI, HENRI DARMON & KARTIK PRASANNA

1

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

ABSTRACT

We give evidence for the refined version of the Bloch-Beilinson conjecture involving coniveau filtrations, by studying several infinite families of CM motives (indexed by the integers r ≥ 1) that are irreducible of Hodge type (2r + 1, 0) + (0, 2r + 1) and whose L-functions vanish at the center. In each case, we construct a corresponding algebraic cycle that is homologically trivial but nontrivial in the top graded piece for the coniveau filtration. Consequently, we obtain new explicit examples of cycles that are nontorsion in the Griffiths group. The proof of the main result relies crucially on p-adic Hodge theoretic methods, and in particular on the relation between images of homologically trivial cycles under the p-adic Abel-Jacobi map and the values of p-adic L-functions.

Contents

Introduction 2 1. Cycles and the Griffiths group 5 2. Coniveau and the refined Bloch-Beilinson conjecture 7 2.1. The refined conjecture 7 2.2. Abel-Jacobi 8 2.3. The Bloch-Kato exponential and logarithm 8 2.4. Coniveau and the Hodge filtration 9 3. p-adic L-functions 12 3.1. The Katz p-adic L-function 12 3.2. p-adic Rankin L-series 12 3.3. Factorization of the p-adic Rankin L-series 13 4. Generalized Heegner cycles and coniveau 14 4.1. Generalized Heegner cycles 14 4.2. Fields of definition 17 4.3. Cycles associated with Grossencharacters 18 5. Miscellany 21 5.1. A non-CM example 21 5.2. The Fermat quartic revisited 24 Appendix A. Tables 28 Appendix B. Integrality of modular forms via intersection theory

by Brian Conrad 30

B.1. Introduction 30 B.2. Intersection-theoretic bounds 32 B.3. Explicit formulas 41 References 46

1Supported partially by NSF grants DMS-1160720 and DMS-0854900.

2Supported partially by NSF grant DMS-0917686.

Introduction

The aim of this article is to apply the main result of [BDP1] to questions surrounding the Beilinson-Bloch (henceforth, BB) conjecture ([Be],[Bl-1]) in the theory of algebraic cycles, leading to the construction of explicit varieties over number fields with non-torsion Griffiths groups. The cycles that we construct correspond to the vanishing at the central point of the L-functions of certain motives^{3}, as predicted by the BB conjectures. The particular motives studied in this article are the simplest interesting ones to which these conjectures apply, namely those attached to Hecke characters of an imaginary quadratic field. In these examples, we construct a homologically trivial cycle that is not just nonzero in the Chow group (with Q-coefficients) but that is also nonzero in its top graded piece for the coniveau filtration, and consequently nontorsion in the Griffiths group.

For X a proper smooth variety over a field k, let CH^{j}(X) denote the Chow group of codimension j cycles modulo rational equivalence, and CH^{j}(X)_{0} the subgroup of classes represented by cycles that are homologically trivial. The group CH^{j}(X)_{0,Q} := CH^{j}(X)_{0} ⊗ Q is equipped with a decreasing coniveau filtration ([Bl-2], [Jan]), denoted N^{i} CH^{j}(X)_{Q}. The definition of this filtration is recalled in section 2.1 below, but for the purposes of the introduction we remind the reader that

N^{0} CH^{j}(X)_{Q} = CH^{j}(X)_{0,Q}, N^{j} CH^{j}(X)_{Q} = 0,

and

N^{j−1} CH^{j}(X)_{Q} = CH^{j}(X)_{alg} ⊗ Q,

where CH^{j}(X)_{alg} is the subgroup of cycles that are algebraically equivalent to zero. Finally, the jth Griffiths group of X is:

Gr^{j}(X) := CH^{j}(X)_{0}/ CH^{j}(X)_{alg}.

To illustrate our main results, let K be an imaginary quadratic field, and suppose that K has class number 1, odd discriminant −D and that the only roots of unity in K are ±1. There is a finite list of such K, namely those with D ∈ {7, 11, 19, 43, 67, 163}. For any positive integer r, let W_{r} denote the Kuga-Sato variety of dimension r + 1 over the modular curve X_{1}(D). Thus W_{r} is a nonsingular projective variety defined over Q and fibered over X_{1}(D), the fibers being isomorphic to the rth power of a varying elliptic curve. Let A be an elliptic curve over Q with complex multiplication by the maximal order of K and (minimal) conductor D^{2}. For any positive integer r, let X_{2r−1,1} be the variety over Q of dimension 2r + 1 given by

X_{2r−1,1} := W_{2r−1} × A.

In Sec. 4.1, we construct (extending the construction in [BDP1]) generalized Heegner cycles

Δ̃_{2r−1,1}^{+}, Δ̃_{2r−1,1}^{−} ∈ CH^{r+1}(X_{2r−1,1}_{/K})_{0} ⊗ Q,

that are defined over K and that are respectively in the ±1-eigenspaces for the action of complex conjugation. Roughly, the construction is based on the fact that a copy of A^{2r−1} occurs as a fiber of W_{2r−1} over a CM point on X_{1}(D); writing this fiber as A′^{2r−1} with A′ ≃ A, these cycles are obtained by applying certain projectors to the cycle

(diagonal ) × (graph of \sqrt{−D})^{r−1} ⊂ (A′ × A) × (A′ × A′)^{r−1} = A′^{2r−1} × A ⊂ W_{2r−1} × A,

and taking a suitable trace. The following theorem is a simple but non-trivial consequence of our main result, formulated in Theorem 4.3.1. (cf. Cor. 4.3.5 and its proof.)

Theorem 1. Suppose that r ∈ Z^{≥1} is odd if D = 7 and is even if D = 11, 19, 43, 67, 163. Then Δ̃_{2r−1,1}^{+} and Δ̃_{2r−1,1}^{−} are both nonzero in the top graded piece of the coniveau filtration, i.e., the classes

[Δ̃_{2r−1,1}^{+}], [Δ̃_{2r−1,1}^{−}] ∈ N^{0} CH^{r+1}(X_{2r−1,1}_{/K})_{Q}/N^{1} CH^{r+1}(X_{2r−1,1}_{/K})_{Q}

are nonzero. In particular, these cycles are nonzero in Gr^{r+1}(X_{2r−1,1}_{/K}) ⊗ Q.

3We use the word motives to mean compatible systems of realizations.

In other words, the theorem states that there is no proper closed subscheme Y of X_{2r−1,1} such that either Δ̃_{2r−1,1}^{+} or Δ̃_{2r−1,1}^{−} is rationally equivalent to a cycle that is supported on Y and is homologically trivial on Y. This should be viewed as giving evidence for a refined version of the BB conjecture (see [Bl-2] and Sec. 2 below) relating cycles to L-functions. We now briefly explain this connection.

The field K of class number one has attached to it a so-called canonical Hecke character ψ of type (1, 0). The character ψ has conductor 𝔡_{K} := (\sqrt{−D}), namely it satisfies

ψ((α)) = α, if α ≡ 1 mod 𝔡_{K}.

In fact, ψ is just the Hecke character associated with the elliptic curve A by the theory of complex multiplication. Thus there is an equality of L-functions:

L(A, s) = L(ψ, s) = L(θ_{ψ}, s),

where θ_{ψ} := ∑_{𝔞} ψ(𝔞)e^{2πiN𝔞·z} ∈ S_{2}(Γ_{0}(D^{2})) (the sum being over integral ideals in 𝒪_{K}) is the modular form of weight 2 associated with A. Let us consider positive integral powers ψ^{t} of the Hecke character ψ and the associated L-functions L(ψ^{t}, s). We are interested in the case where L(ψ^{t}, s) is self-dual and such that the center of its functional equation is an integer, so that the BB conjecture applies at this point. This occurs exactly when t is odd, so we write t = 2r + 1, with r a non-negative integer. An easy computation using the fact that ψ^{2} is an unramified Hecke character of infinity type (2, 0) shows that:

(I-1)  sign L(ψ^{2r+1}, s) = \{ \begin{matrix}sign L(ψ, s), if r is even; \\ − sign L(ψ, s), if r is odd. \end{matrix}

As for the sign of L(ψ, s), it so happens that

(I-2)  sign L(ψ, s) = \{ \begin{matrix}+1, if D = 7; \\ −1, if D ∈ \{11, 19, 43, 67, 163\}. \end{matrix}

Let f be the modular form

f = θ_{ψ^{2r}} = ∑_{𝔞} ψ^{2r}(𝔞)e^{2πiN𝔞·z} ∈ S_{2r+1}(Γ_{0}(D), ε_{K}),

where ε_{K} is the quadratic Dirichlet character of conductor D attached to the extension K/Q. By [Scholl], the form f contributes to the cohomology of the Kuga-Sato variety W_{2r−1}, in fact to the middle cohomology H^{2r}(W_{2r−1}). By the Künneth formula, the motive H^{2r+1}(X_{2r−1,1}) then contains

(I-3)  M_{f} ⊗ M_{θ_{ψ}} = M_{θ_{ψ^{2r}}} ⊗ M_{θ_{ψ}} = M_{θ_{ψ^{2r+1}}} ⊕ M_{θ_{ψ^{2r−1}}} (−1)

as a submotive, where we write M_{f} and M_{θ_{ψ^{i}}} for the motives over Q attached to f and θ_{ψ^{i}} respectively. The BB conjectures then imply, on account of (I-1) and (I-2), that

rank CH^{r+1}(X_{2r−1,1})_{0,Q} ≥ 1

for odd values of r when D = 7 and for even values of r if D ≠ 7. Moreover, since M_{θ_{ψ^{2r+1}}} has Hodge type (2r + 1, 0) + (0, 2r + 1), the refined version of the BB conjecture (recalled below in Sec. 2.1) implies that top graded piece of this Chow group for the coniveau filtration is nontrivial. Theorem 1 above thus provides evidence for this conjecture by constructing an explicit nonzero element Δ̃_{2r−1,1}^{+} in the quotient CH^{r+1}(X_{2r−1,1})_{0,Q}/N^{1} CH^{r+1}(X_{2r−1,1})_{Q}. In fact, it does somewhat more, namely construct also a similar element Δ̃_{2r−1,1}^{−} over K that is in the minus-eigenspace for complex conjugation; this is also predicted by the behavior of the L-function, since the L-function of M_{θ_{ψ^{2r+1}}} over K is simply the square of its L-function over Q.

We now say a few words about the proof of Theorem 1. The content of this theorem is that for each imaginary quadratic field K above, there exists an explicit infinite family of varieties ℱ_{K} of increasing dimension (or more precisely, an infinite family of irreducible motives) for which a part of the refined BB conjecture can be verified. The main ingredient in the proof is the relation between p-adic Abel-Jacobi images of cycles and p-adic L-functions that is proved in [BDP1]. This relation can be used to reduce the validity of Theorem 1 for any given variety X in ℱ_{K} to the nonvanishing of the Katz p-adic L-function associated with K at a specific point outside the range of interpolation that is determined by the dimension of X. If the value in question turns out to be a p-adic unit, then by the yoga of p-adic L-functions, this nonvanishing mod p, and therefore the validity of the theorem, propagates to all the varieties in ℱ_{K} whose dimension lies in the same congruence class as dim(X) mod 2(p − 1). Happily, it turns out that we can find for each K in our list above, a suitable prime p such that this nonvanishing mod p holds for all the relevant congruence classes mod 2(p − 1), thereby reducing the theorem, which applies to an infinite set of varieties of unbounded dimension, to a finite (and simple!) computation.

Remark 2. In the main text, we prove a somewhat more general result, namely for any i such that 0 ≤ i ≤ r − 1 and i ≡ r − 1 mod 2, we construct cycles Δ̃_{r+i,r−i}^{±} on the variety

X_{r+i,r−i} := W_{r+i} × A^{r−i},

and show that these are nontrivial in N^{0}/N^{1}. This is the content of Cor. 4.3.5 below.

Remark 3. Theorem 1 suggests that for the Hecke characters ψ^{2r+1} considered above, whenever the sign of the functional equation is −1, the L-function vanishes to order exactly one, i.e,

(I-4)  If sign L(ψ^{2r+1}, s) = −1, then L′(ψ^{2r+1}, r + 1) ≠ 0.

We do not know how to prove this at this point. In fact, (I-4) would follow if one could

(i) Prove a precise relation between the (Arakelov) height of the cycle Δ̃_{2r−1,1}^{±} and the derivative L′(ψ^{2r+1}, r + 1), and

(ii) If one knew the non-degeneracy of the height pairing, at least on the lines spanned by Δ̃_{2r−1,1}^{±}. Of these, (i) is certainly within reach by the methods of [Zh], but (ii) would seem to require a genuinely new idea. It would be very interesting therefore to prove (I-4) directly, say by purely analytic means.

The reader might wonder what happens in the complementary cases, namely r even when D = 7 and r odd otherwise. In these cases,

sign L(ψ^{2r+1}, s) = +1,  while  sign L(ψ^{2r−1}, s) = −1.

Since M_{θ_{ψ^{2r−1}}} has Hodge type (2r − 1, 0) + (0, 2r − 1), one expects that the projection of the cycle Δ̃_{2r−1,1}^{±} on the (f, θ_{ψ})-component lands in N^{1} CH^{r+1}(X_{2r−1,1})_{Q} but is nonzero in the quotient N^{1}/N^{2}. A weaker statement is that one can apply some projector to Δ̃_{2r−1,1}^{±} to yield a cycle with this property. However, it seems very hard to construct such a projector since this is essentially equivalent to verifying an open case of the Tate conjecture. The same difficulty persists in the more general situation of Remark 2. Consider for example the extreme case i = 0 in Remark 2, so that r is odd and D ≠ 7 and the relevant L-function is

L(M_{θ_{ψ^{r+1}}} ⊗ M_{θ_{ψ^{r}}} , s) = L(ψ^{2r+1}, s)L(ψ, s − r).

Since

sign L(ψ^{2r+1}, s) = +1, while sign L(ψ, s) = −1,

and M_{θ_{ψ}} has Hodge type (1, 0) + (0, 1), one should expect that the (θ_{ψ^{r+1}}, θ_{ψ^{r}})-component of Δ̃_{r,r}^{±} lands in N^{r} CH^{r+1}(X_{r,r}), that is, comes by correspondence from a divisor of degree zero on a curve. Since L(ψ, s) is the L-function of the elliptic curve A, such a curve should naturally contain A as a factor in its Jacobian and one therefore gets from Δ̃_{r,r}^{±} a (conjectural) construction of rational points on A. This idea is explored in detail in [BDP3], which formulates such a conjectural motivic construction precisely and shows (unconditionally) that its realization in p-adic étale cohomology yields classes that come from rational points. On the other hand, its realization in Betti cohomology yields points on A(C) that can be experimentally verified to be rational in many cases, but which cannot be proved to be rational except in a handful of low dimensional examples where the relevant case of the Tate conjecture can be verified by hand.

Remark 4. It may be worth pointing out here the analogy with the Gross-Zagier theorem, in which in certain instances the Rankin-Selberg motive being considered splits as a sum of rank-2 motives and the associated L-function factors as a product of two degree-2 L-functions. The sign of the degree-4 L-function is −1 and the signs of the two individual factors determine the action of complex conjugation on the cycle. For example, the simplest case of such a factorization is

L(E_{/K}, s) = L(E, s) · L(E, ε_{K}, s),

where E is an elliptic curve over Q, and the signs of the individual factors determine whether the corresponding Heegner point is defined over K or Q. In our case, again we have such a factorization but the individual signs seem to determine a somewhat more sophisticated invariant of the cycle, namely its exact position in the coniveau filtration.

The paper ends with two miscellaneous but related results. The first (in Sec. 5.1) treats similar phenomena for some non-CM motives. The results here use an explicit calculation of the relevant non-CM form (and of a derivative of this form) at a CM point, and in particular that the algebraic parts of these computed values are p-units for a suitably chosen prime p. This requires identifying the computed values as algebraic numbers, and here we make essential use of Appendix B by Brian Conrad that provides integrality results for modular forms in terms of the q-expansion at the cusp at infinity. The second miscellaneous result is a new proof by the methods of this paper of Bloch’s theorem that the Chow group of codimension-2 cycles on the Jacobian of the Fermat quartic over Q contains a cycle (the Ceresa cycle) that is homologically trivial but non-torsion in the Griffiths group. This result is of historical importance since it provided the first concrete evidence for the generalization of the Birch-Swinnerton-Dyer conjecture to the BB conjecture. The following section describes some of this history and the relation between Bloch’s method and the one of this article.

1. Cycles and the Griffiths group

Let X be a smooth projective variety over C of dimension n. For 1 ≤ j ≤ n, let

cl : CH^{j}(X) → H^{2j}(X, C)

denote the cycle class map, CH^{j}(X)_{0} its kernel (the group of homologically trivial cycles) and

AJ : CH^{j}(X)_{0} → J^{j}(X)

the Abel-Jacobi map into the jth Griffiths intermediate Jacobian of X. Recall that J^{j}(X) is the intermediate Jacobian of the Hodge structure H^{2j−1}(X), i.e.,

J^{j}(X) := (Fil^{j} H^{2j−1}(X, C))^{∨}/H_{2j−1}(X, Z) = (⊕_{i≥j}H^{i,2j−1−i}(X)^{∨})/H_{2j−1}(X, Z).

In the extreme cases j = 1 and j = n, J^{j}(X) is an abelian variety, namely the identity component of the Picard variety of X and, respectively, the Albanese of X. In general, J^{j}(X) is only a complex torus. The maximal abelian subvariety J^{j}(X)_{alg} of J^{j}(X) is defined to be the complex subtorus of J^{j}(X) which is the intermediate Jacobian of the largest sub-Hodge structure of H^{j,j−1}(X) ⊕ H^{j−1,j}(X). This terminology is motivated by the fact that the image of CH^{j}(X)_{alg} (the subgroup of CH^{j}(X)_{0} consisting of cycles algebraically equivalent to zero) under AJ is a complex subtorus of J^{j}(X) that is contained in J^{j}(X)_{alg} and has the structure of an abelian variety, and further, the Hodge conjecture implies that this image is equal to J^{j}(X)_{alg}. The jth Griffiths group Gr^{j}(X) is the quotient

Gr^{j}(X) := CH^{j}(X)_{0}/ CH^{j}(X)_{alg}.

This quotient is at most countable, as can be shown using the theory of Hilbert schemes; in particular, this implies that AJ has no chance of being surjective if Fil^{j} H^{2j−1}(X) > H^{j,j−1}(X). In fact, for j = n and j = 1, the group Gr^{j}(X) is always zero since algebraic and homological equivalence coincide for 0-cycles (trivially) and for divisors by the Lefschetz (1, 1)-theorem. However, in contrast to these cases, for other values of j, it was only in the late 60’s that Griffiths showed in an important series of articles [Gr-1,2] that Gr^{j}(X) can be nonzero and indeed even nontorsion. The particular example that Griffiths studied was Gr^{2} of the very general hypersurface of degree 5 in ℙ^{4}. Griffiths’ work was later generalized by Clemens [Cl] who showed for the same example that Gr^{2}(X) is not even finitely generated. Yet another example is due to Ceresa [Cer], who proved that for C the very general curve of genus g ≥ 3, the Ceresa cycle C − [−1]^{∗}C in X = Jac(C) is homologically trivial and nontorsion in Gr^{g−1}(X). However, the arguments in all these cases made essential use of transcendental elements in the base field and assumed the variety is very general, that is, lies outside of a countable union of proper locally closed subsets of the parameter space. This left open the question of giving an explicit example of such a phenomenon and also whether the Griffiths group could be nontorsion for a variety defined over a number field. These questions were settled by work of Bruno Harris and Spencer Bloch, and led to the first nontrivial evidence for the BBconjecture relating the behaviour of L-functions at the central point to algebraic cycles. We will quickly recall their work since the motives we deal with later in this article are generalizations of exactly the examples considered by Harris and Bloch.

Let C be the Fermat quartic:

T_{0}^{4} + T_{1}^{4} = T_{2}^{4}.

Then C is a curve of genus 3 defined over Q. Let X be the Jacobian of C. Choose an embedding of C in X and let

Ξ = [C] − [(−1)^{∗}C] ∈ Gr^{2}(X).

The class of Ξ in Gr^{2}(X) is independent of the choice of embedding of C in X. Harris [Har] proved that in fact Ξ is non-zero in Gr^{2}(X) by showing that its image under AJ is not contained in J^{2}(X)_{alg}. Since X is a threefold, this is equivalent to showing the following: writing Ξ = ∂Γ for a singular 3-chain Γ, the functional

H^{3,0}(X) → C, ω ↦ ∫_{Γ} ω

is not in the image of H_{3}(X, Z) in H^{3,0}(X)^{∨}. In this special case it turns out that the image of H_{3}(X, Z) is a lattice (of periods) in H^{3,0}(X)^{∨} (since X is isogenous to a product of three elliptic curves with complex multiplication by the ring of integers of the same imaginary quadratic field) and the problem can therefore be reduced to showing that the ratio of a certain iterated integral to a period (see [Har], equation (1)) is not an integer. This in turn can be verified by a numerical computation; in fact, if one could show that this ratio (which experimentally is 1.24178...) is not rational then it would follow that Ξ is even non-torsion in Gr^{2}(X). Such an approach to showing Ξ is non-torsion however seems out of reach even at this point of time.

The reasons to expect Ξ to be non-torsion in Gr^{2}(X) come from a refined version of the Bloch-Beilinson conjecture. We will now recall what the relevant L-functions are. The abelian variety X is isogenous over Q to the product E × E × E′, where

E : y^{2} = x^{3} − x, E′ : y^{2} = x^{3} + x.

The curves E and E′ have conductors 32 and 64 respectively, and have CM by the ring of integers of the imaginary quadratic field K = Q(i). Let ψ and ψ′ be the associated Grössencharacters of K of type (1, 0). These have the property that their central characters ε_{ψ}, ε_{ψ′} (that is, their restrictions to the ideles of Q) equal ε_{K}, the quadratic Dirichlet character associated with the extension K/Q. Further, ψ′ = ψμ, for a certain quadratic character μ of K, namely that attached to the quadratic extension Q(ζ_{8})/K.

For any Hecke character χ of K, we write M_{θ_{χ}} for the motive over Q of rank 2 whose associated Galois representation is Ind_{Q}^{K}(χ). We also write V_{ℓ}(M_{θ_{χ}}) for the ℓ-adic realization of M_{θ_{χ}}. On account of the extra complex multiplications, the motive H^{3}(X) splits as a sum I ⊕ M where h^{3,0}(I) = 0 and M is of type (3, 0) + (0, 3). Indeed, I is just a sum of three copies of M_{θ_{ψ′}} (−1), while M = M_{θ_{χ}}, where χ := ψ^{2}ψ′ = ψ^{3}μ. Thus

L(H^{3}(X), s) = L(ψ′, s − 1)^{3} · L(χ, s).

It turns out in this case that the central point is s = 2, and L(ψ′, 1) ≠ 0, while L(χ, s) vanishes to order 1 at s = 2. The BB conjecture then predicts that the rank of CH^{2}(X)_{0} is ≥ 1. In addition, since M_{θ_{χ}} is purely of type (3, 0) + (0, 3), one might expect moreover that the rank of the quotient Gr^{2}(X) is also ≥ 1. This expectation is made precise in the refined version of the BB conjecture involving coniveau filtrations, recalled below. That Bloch [Bl-1] was able to verify this expectation in this case was for him the first nontrivial evidence for the BB conjecture^{4} in higher dimensions.

Bloch’s method is to study the map

CH^{2}(X)_{0} ⟶^{AJ_{ét}} H^{1}(\overline{Q}/K, H^{3}(X\overline{Q}, Q_{ℓ}(2)) ⟶^{P_{∗}} H^{1}(\overline{Q}/K, V_{ℓ}(M_{θ_{χ}}(2))),

where AJ_{ét} is the étale Abel-Jacobi map, and P_{∗} is induced by a projector P that cuts out the motive M_{θ_{χ}}. (Such a projector exists on account of the extra complex multiplications.) Let γ_{ℓ} := AJ_{ét}(Ξ) and β_{ℓ} := P_{∗}(γ_{ℓ}). Also, let p ≠ ℓ be a prime at which X has good reduction. By restricting β_{ℓ} to a decomposition group at a prime above p and using a clever geometric construction in characteristic p, Bloch is able to show that β_{ℓ} ≠ 0, and this implies (for instance, using Prop. 2.2.1 below) that Ξ is nontorsion in Gr^{2}(X). This idea was extended by Schoen [Schoen1] and combined with a calculation of the complex Abel-Jacobi map to show that if W is the Kuga-Sato threefold over the modular curve X(N), N ≥ 3, then the Griffiths group of X over \overline{Q} is not finitely generated. There is no connection with L-functions made in loc. cit., however, another article of Schoen [Schoen2] studies the BB conjecture for a specific Kuga-Sato threefold over an infinite collection of number fields and provides some evidence in support of the conjecture.

4In fact, Bloch does not call it a conjecture at all but rather a recurrent fantasy, ranking it above idle speculation on account of this one example !

Our approach to constructing examples of varieties over number fields with non-torsion Griffiths groups is based on Bloch’s idea of using the étale Abel-Jacobi map. A key difference is that we study the restriction of β_{ℓ} to a decomposition group D_{p} at a prime above p in the case ℓ = p rather than ℓ ≠ p. Via the comparison theorems of p-adic Hodge theory, the restriction of β_{p} (resp. γ_{p}) to D_{p} can be viewed as a linear form on Fil^{0} D_{dR}(V_{θ_{χ}}(2)) (resp. on Fil^{0} D_{dR}(H^{3}(X\overline{Q}, Q_{p}(2))) = Fil^{2} H_{dR}^{3}(X_{Q_{p}})), and to show that Ξ is nontorsion in Gr^{2}(X), it suffices to show that β_{p} is not zero (resp. that γ_{p} is not zero on H^{3,0}(X) ⊂ Fil^{2} H_{dR}^{3}(X, Q_{p}).) In this way, the method begins to resemble that of Harris, except with the usual Abel-Jacobi map replaced by the p-adic Abel-Jacobi map. However, the p-adic Abel-Jacobi map has the distinct advantage that its target is torsion-free, since there are no periods (in a naive sense) to deal with! Further, it turns out that in the p-adic case, there are precise relations between the coordinates of the Abel-Jacobi images in suitable bases and the values of certain p-adic L-functions at points outside the range of interpolation. The main theorem of [BDP1] provides an instance of this relation for certain Rankin-Selberg motives, which can be specialized to yield motives of the sort considered above. In fact, there are already tenuous hints of such a relation for the motives attached to CM Hecke characters in the second article of Bloch [Bl-2], where he proves a Coates-Wiles type theorem showing that the nonvanishing of the restriction of the p-adic AJ-class of a homologically trivial cycle to a decomposition group at p implies the vanishing of a central L-value. Our results are in the opposite direction, namely, we show that the vanishing of an L-value (for sign reasons) corresponds (via the nonvanishing of a p-adic L-value) to the nonvanishing of the p-adic AJ class of an explicit homologically trivial cycle.

2. Coniveau and the refined Bloch-Beilinson conjecture

In this section alone, we let k denote a number field, while K will denote the completion of k at a finite place above the rational prime p. This will allow us to import notations from p-adic Hodge theory more easily.

2.1. The refined conjecture. We recall, following [Bl-2] §1, the refined version of the BB conjecture involving coniveau filtrations. Let G_{k} := Gal(k̄/k) and X a smooth projective variety over k of dimension n. Let CH^{j}(X) (resp. CH^{j}(X)_{0}) denote the Chow group of cycles (resp. homologically trivial cycles) of codimension j on X defined over k modulo rational equivalence. The BB conjecture predicts that for any 1 ≤ j ≤ n, CH^{j}(X)_{0} is a finitely generated abelian group and

dim_{Q} CH^{j}(X)_{0,Q} = ord_{s=j} L(H^{2j−1}(Xk̄), s).

For the refined version, we need to recall the definitions of the coniveau filtrations. On H^{∗}(Xk̄, Q_{ℓ}), this is the Galois stable filtration defined by

N^{i}H^{∗}(Xk̄, Q_{ℓ}) := lim_{codim Y ≥i^{Y ⊂Xk̄}} ker(H^{∗}(Xk̄, Q_{ℓ}) → H^{∗}(Xk̄ − Y, Q_{ℓ})),

the limit being taken over closed subschemes Y ⊂ Xk̄ of codimension ≥ i. Define N^{i} CH^{j}(X)_{Q} to consist of those classes γ ∈ CH^{j}(X)_{Q} satisfying the following property: there exists a cycle c representing γ and a closed subscheme Y ⊂ Xk̄ of codimension ≥ i such that supp(c) ⊂ Y and c is homologous to zero on Y. Thus N^{0} CH^{j}(X)_{Q} = CH^{j}(X)_{0,Q} and N^{j} CH^{j}(X)_{Q} = 0. Further, N^{j−1} CH^{j}(X)_{Q} = CH^{j}(X)_{alg,Q} (see [Bl-2], Lemma 1.1), hence

Gr^{j}(X)_{Q} = N^{0} CH^{j}(X)_{Q}/N^{j−1} CH^{j}(X)_{Q}.

Conjecture 2.1.1. ([Bl-2], p. 381) Let gr_{N}^{i}H^{∗} := N^{i}H^{∗}(Xk̄, Q_{ℓ})/N^{i+1}. Then L(gr_{N}^{i}H^{∗}, s) is independent of ℓ for any value of ∗, and

dim_{Q} gr_{N}^{i} CH^{j}(X)_{Q} = ord_{s=j} L(gr_{N}^{i}H^{2j−1}, s).

2.2. Abel-Jacobi. The conjecture above is motivated by studying the ℓ-adic Abel-Jacobi map:

AJ_{ℓ} : CH^{j}(X)_{0,Q} → H^{1}(G_{k}, H^{2j−1}(Xk̄, Q_{ℓ}(j))),

defined as in [Nek1], §1, for example, and its relation to the coniveau filtration. The following suggestive proposition in that direction is due to Bloch ([Bl-2], Prop. 1.5.)

Proposition 2.2.1. The map AJ_{ℓ} is compatible with the coniveau filtrations and induces natural maps

gr_{N}^{i} CH^{j}(X)_{Q} → H^{1}(G_{k}, gr_{N}^{i}H^{2j−1}(Xk̄, Q_{ℓ}(j))).

This proposition implies there are natural maps δ_{i} for 0 ≤ i ≤ j such that the diagram below commutes. (Note that the vertical arrows on the right are inclusions since N^{i}H^{2j−1}(Xk̄, Q_{ℓ}(j))/N^{i+1} is pure of weight −1, hence has no nonzero G_{k}-invariants.)

(2.2.1)  \xymatrix{CH^{j}(X)_{0,Q} \ar[r]^{δ_{0}=AJ_{ℓ}} & H^{1}(G_{k}, H^{2j−1}(Xk̄, Q_{ℓ}(j))) \\ N^{i} CH^{j}(X)_{0,Q} \ar[u]^{∪} \ar[r]^{δ_{i}} & H^{1}(G_{k}, N^{i}H^{2j−1}(Xk̄, Q_{ℓ}(j))) \ar[u]^{∪} \\ N^{i+1} CH^{j}(X)_{0,Q} \ar[u]^{∪} \ar[r]^{δ_{i+1}} & H^{1}(G_{k}, N^{i+1}H^{2j−1}(Xk̄, Q_{ℓ}(j))) \ar[u]^{∪}}

Suppose that v is a prime of k at which X has good reduction, lying over the rational prime p. Fix a place v̄ of k̄ above v and let G_{v} denote the decomposition group of v̄ over v. Take ℓ = p, and consider the composite map:

(2.2.2)  δ_{0,v} : CH^{j}(X)_{Q} ⟶^{δ_{0}} H^{1}(G_{k}, H^{2j−1}(Xk̄, Q_{p}(j))) ⟶^{res_{v}} H^{1}(G_{v}, H^{2j−1}(Xk̄, Q_{p}(j))).

We will need to study the image of the coniveau filtration under the map δ_{0,v}, for which we need some notations and results from p-adic Hodge theory, recalled in the following section.

2.3. The Bloch-Kato exponential and logarithm. We summarize some results from p-adic Hodge theory, referring the reader to [BK], [Nek2] for more details. Let K be a finite extension of Q_{p} and G_{K} := Gal(\overline{K}/K). We denote by K_{0} the maximal unramified extension of Q_{p} in K and by σ the absolute Frobenius acting on K_{0} so that σ induces the pth power map on the residue field of K_{0}. Let V be a p-adic representation of G_{K}, i.e., a finite dimensional Q_{p}-vector space with a continuous action of G_{K}. Associated with V are certain vector spaces (over K_{0} or K) with additional structures:

(i) D_{cris}(V) := (V ⊗_{Q_{p}} B_{cris})^{G_{K}}, a K_{0}-vector space equipped with a σ-semilinear bijective map

φ : D_{cris}(V) → D_{cris}(V).

(ii) D_{st}(V) := (V ⊗_{Q_{p}} B_{st})^{G_{K}}, a K_{0}-vector space equipped with a σ-semilinear bijective map φ : D_{st}(V) → D_{st}(V) and a K_{0}-linear map N : D_{st}(V) → D_{st}(V) (called the monodromy operator)

satisfying Nφ = pφN. We let φ_{K} denote the K_{0}-linear map φ^{[K_{0}:Q_{p}]}.

(iii) D_{dR}(V) := (V ⊗_{Q_{p}} B_{dR})^{G_{K}}, a K-vector space equipped with a decreasing filtration

(Fil^{r} D_{dR}(V))_{r∈Z},
satisfying ∪ Fil^{r} D_{dR}(V) = D_{dR}(V), and ∩ Fil^{r} D_{dR}(V) = 0.

These are related by:

• D_{cris}(V) = D_{st}(V)^{N=0}.

• The canonical maps

(2.3.1)  D_{cris}(V) ⊗_{K_{0}} K → D_{dR}(V)

and

(2.3.2)  D_{st}(V) ⊗_{K_{0}} K → D_{dR}(V)

are injective.

• dim_{K_{0}} D_{cris}(V) ≤ dim_{K_{0}} D_{st}(V) ≤ dim_{K} D_{dR}(V) ≤ dim_{Q_{p}}(V).

The representation V is said to be crystalline (resp. semistable, resp. de Rham) if dim_{K_{0}} D_{cris}(V) (resp. dim_{K_{0}} D_{st}(V), resp. dim_{K} D_{dR}(V)) is equal to dim_{Q_{p}}(V). Thus V crystalline ⇒ V semistable ⇒ V de Rham. Further, if V is semistable, then the map (2.3.2) is an isomorphism, while if V is crystalline, then N = 0 on D_{st}(V), D_{cris}(V) = D_{st}(V) and both (2.3.1) and (2.3.2) are isomorphisms.

Finally, we need to recall certain subspaces of H^{1}(G_{K}, V). The subspace H_{f}^{1}(G_{K}, V) is defined by:

H_{f}^{1}(G_{K}, V) = ker(H^{1}(G_{K}, V) → H^{1}(G_{K}, V ⊗ B_{cris})).

Likewise, the subspaces H_{st}^{1}(G_{K}, V) and H_{g}^{1}(G_{K}, V) are defined by replacing B_{cris} by B_{st} and B_{dR} respectively. Concretely, if V is a crystalline (resp. semistable, resp. de Rham) representation, then an element ξ of H_{f}^{1}(G_{K}, V) (resp. H_{st}^{1}(G_{K}, V), resp. H_{g}^{1}(G_{K}, V)) is represented by an extension of p-adic representations:

(2.3.3)  0 → V → V′ → Q_{p} → 0,

where V′ is also crystalline (resp. semistable, resp. de Rham). It is known that if V is semistable, then H_{st}^{1}(G_{K}, V) = H_{g}^{1}(G_{K}, V) ([Nek2], Prop. 1.24), though we will not need this fact below.

Suppose that V is a de Rham representation satisfying D_{cris}(V)^{φ_{K}=1} = 0. Then the Bloch-Kato exponential ([BK], Defn. 3.10 and Thm. 4.1(ii)) gives an isomorphism:

exp : D_{dR}(V)/ Fil^{0} D_{dR}(V) ≃ H_{f}^{1}(G_{K}, V).

The inverse isomorphism will be denoted

log : H_{f}^{1}(G_{K}, V) ≃ D_{dR}(V)/ Fil^{0} D_{dR}(V).

If further V is crystalline, the log map has the following alternate description which is very useful. Namely, let ξ ∈ H_{f}^{1}(G_{K}, V) and suppose ξ is represented by an extension as in (2.3.3) above, so that V′ is also crystalline. Applying the functor D_{dR} yields exact sequences:

[figure: 0; D_{dR}(V); D_{dR}(V′); K; 0; 0; Fil^{0} D_{dR}(V); Fil^{0} D_{dR}(V′); K; 0]

Pick elements x ∈ Fil^{0} D_{dR}(V′), y ∈ D_{cris}(V′)^{φ_{K}=1} ⊂ D_{dR}(V′) such that both map to the element 1 ∈ K. Then x−y ∈ D_{dR}(V) and its class in D_{dR}(V)/ Fil^{0} D_{dR}(V) is well defined since D_{cris}(V)^{φ_{K}=1} = 0. The alternate description of log(ξ) is then:

(2.3.4)  log(ξ) = [x − y] ∈ D_{dR}(V)/ Fil^{0} D_{dR}(V).

2.4. Coniveau and the Hodge filtration. We now apply the tools from the previous section to study the map δ_{0,v} defined in (2.2.2) above. Let V = H^{2j−1}(Xk̄, Q_{p}(j)), K = k_{v} and identify G_{v} with G_{K}. By [Nek1], Thm 3.1 (i), the image of δ_{0,v} is contained in H_{f}^{1}(G_{K}, V). Since V is a de Rham representation (in fact even crystalline) and D_{cris}(V)^{φ_{K}=1} = 0 (by [KMe]), the log map gives an isomorphism

log : H_{f}^{1}(G_{K}, V) ≃ D_{dR}(V)/ Fil^{0} D_{dR}(V).

As a filtered vector space, D_{dR}(V) is isomorphic to H_{dR}^{2j−1}(X_{K}) equipped with the usual Hodge filtration shifted by j. Thus

Fil^{0} D_{dR}(V) = Fil^{j} H_{dR}^{2j−1}(X_{K}).

Further, H_{dR}^{2j−1}(X_{K}) is equipped with a nondegenerate (Poincaré) pairing such that Fil^{i} and Fil^{2j−i} are exact annihilators of each other. Thus there is a canonical isomorphism:

PD : D_{dR}(V)/ Fil^{0} D_{dR}(V) = H_{dR}^{2j−1}(X_{K})/ Fil^{j} H_{dR}^{2j−1}(X_{K}) ≃ (Fil^{j} H_{dR}^{2j−1}(X_{K}))^{∨}.

Finally let us define

α_{v} := log ∘δ_{0,v} : CH^{j}(X)_{0,Q} → H_{dR}^{2j−1}(X_{K})/ Fil^{j} H_{dR}^{2j−1}(X_{K}),

and

β_{v} := PD ∘ log ∘δ_{0,v} : CH^{j}(X)_{0,Q} → (Fil^{j} H_{dR}^{2j−1}(X_{K}))^{∨}.

The following key proposition relates the coniveau filtration on the Chow group to the Hodge filtration on de Rham cohomology.

Proposition 2.4.1. For any integer i in the range 0 ≤ i ≤ j,

α_{v}(N^{i} CH^{j}(X)_{0,Q}) ⊆ Fil^{i} H_{dR}^{2j−1}(X_{K})/ Fil^{j} H_{dR}^{2j−1}(X_{K}).

The following corollary is immediate:

Corollary 2.4.2. For any integer i as above, β_{v}(N^{i} CH^{j}(X)_{0,Q}) annihilates Fil^{2j−i} H_{dR}^{2j−1}(X_{K}). In particular, β_{v}(N^{1} CH^{j}(X)_{0,Q}) annihilates Fil^{2j−1} H_{dR}^{2j−1}(X_{K}) = H^{0}(X_{K}, Ω^{2j−1}).

Proof: (of Prop. 2.4.1) Let γ ∈ N^{i} CH^{j}(X)_{0,Q} and let (Y, c) be a pair consisting of a closed subscheme f : Y ⊂ Xk̄ of codimension i and a cycle c on Xk̄ representing γ such that supp(c) is contained in Y and c is homologous to zero on Y. We need to show that α_{v}(γ) lies in Fil^{i} H_{dR}^{2j−1}(X_{K})/ Fil^{j} H_{dR}^{2j−1}(X_{K}). Replacing γ by an integer multiple, we may assume (as in the proof of [Bl-2] Prop. 1.5) that Y and c are defined over k.

Let us suppose for the moment that Y is smooth over k and has good reduction at v. Writing W := H^{2(j−i)−1}(Yk̄, Q_{p}(j − i)), there is a commutative diagram:

(2.4.1)  \xymatrix{CH^{j−i}(Y)_{0,Q} \ar[d]^{f_{∗}} \ar[r]^{res_{v} ∘ AJ_{Y}} & H_{f}^{1}(G_{K}, W) \ar[d]^{f_{∗}} \ar[r]^{≃^{log}} & D_{dR}(W)/ Fil^{0} = H_{dR}^{2(j−i)−1}(Y_{K})/ Fil^{j−i} \ar[d]^{f_{∗}} \\ CH^{j}(X)_{0,Q} \ar[r]^{δ_{0,v}=res_{v} ∘ AJ_{X}} & H_{f}^{1}(G_{K}, V) \ar[r]^{≃^{log}} & D_{dR}(V)/ Fil^{0} = H_{dR}^{2j−1}(X_{K})/ Fil^{j}}

Since the pushforward (Gysin) map f_{∗} on the far right takes Fil^{·} to Fil^{·+i} (as can be seen by embedding K in C and using Hodge theory), it follows that the image of CH^{j−i}(Y)_{0,Q} under log ∘δ_{0,v} ∘ f_{∗} is contained in Fil^{i} H_{dR}^{2j−1}(X_{K})/ Fil^{j} H_{dR}^{2j−1}(X_{K}), hence α_{v}(γ) is also contained in the same subspace.

In general Y may be singular and in any case may have bad reduction, so we argue somewhat differently. Let Z := supp(c) and write X̄, Ȳ , Z̄ for Xk̄, Yk̄, Zk̄ respectively. Also let ξ_{γ} := δ_{0,v}(γ). Then (as in [Bl-2], Prop. 1.5, again) the Abel-Jacobi map (restricted to the place v) associates with c an element

ξ_{c} ∈ H^{1}(G_{K}, H2j−1 ; Ȳ(X̄)(j))

which maps to ξ_{γ} under the natural map

H^{1}(G_{K}, H2j−1 ; Ȳ(X̄)(j)) → H^{1}(G_{K}, H^{2j−1}(X̄)(j)).

Both ξ_{γ} and ξ_{c} may be viewed as extensions coming from a long exact sequence of cohomology. Indeed, there is a commutative diagram:

(2.4.2)

[figure: 0 = H2j−1 ; Z̄(X̄, Q_{p}(j)); H^{2j−1}(X̄, Q_{p}(j)); H^{2j−1}(X̄ − Z̄, Q_{p}(j)); H2j ; Z̄(X̄, Q_{p}(j)); 0 = H2j−1 ; Z̄(X̄, Q_{p}(j)); H2j−1 ; Ȳ(X̄, Q_{p}(j)); H_{Ȳ −Z̄}^{2j−1}(X̄ − Z̄, Q_{p}(j)); H2j ; Z̄(X̄, Q_{p}(j))]

and the extensions ξ_{γ} and ξ_{c} are obtained by pulling back the first and second lines by the map

Q_{p} → H2j ; Z̄(X̄, Q_{p}(j)), 1 ↦ cl(c).

(Note that cl(c) maps to 0 in the next term of each row, since c is homologically trivial on Y.) Thus, writing W := H2j−1 ; Ȳ(X̄, Q_{p}(j)), we have a commutative diagram of G_{K}-representations

(2.4.3)  [figure: 0; V; V′; Q_{p}; 0; 0; W; W′; Q_{p}; 0]

where the top row represents ξ_{γ} and the bottom row represents ξ_{c}.

Since V is crystalline and ξ_{γ} ∈ H_{f}^{1}(G_{K}, V), the representation V′ is crystalline. In contrast with V and V′, the representations W and W′ need not be crystalline. Nevertheless, by a theorem of Kisin ([Kis], Thm 3.2), W and W′ are at least potentially semistable, so after making a finite extension L/K, we may assume ξ_{c} is semistable, that is,

res_{K/L} ξ_{c} ∈ H_{st}^{1}(G_{L}, W).

Applying the functors D_{st} and D_{dR} (over L), we see that there are commutative diagrams

[figure: 0; D_{st}(V); D_{st}(V′); L_{0}; 0; 0; D_{st}(W); ι; D_{st}(W′); ι; L_{0}; 0]

and

[figure: 0; D_{dR}(V); D_{dR}(V′); L; 0; 0; D_{dR}(W); ι; D_{dR}(W′); ι; L; 0]

with the rows being exact. Pick elements x′ ∈ Fil^{0} D_{dR}(W′), y′ ∈ D_{st}(W′)^{φ_{L}=1}, such that both x′ and y′ map to the element 1 ∈ L. Let

x = ι(x′), y = ι(y′),

so that

x ∈ Fil^{0} D_{dR}(V′), y ∈ D_{st}(V′)^{φ_{L}=1} = D_{cris}(V′)^{φ_{L}=1},

and x, y both map to the element 1 ∈ L. By the alternate description of the log map, we have

log(res_{K/L}(ξ_{γ})) = [x − y] = [ι(x′) − ι(y′)] = [ι(x′ − y′)]
∈ D_{dR}(V)/ Fil^{0} = H_{dR}^{2j−1}(X_{L})/ Fil^{j} H_{dR}^{2j−1}(X_{L}).

But x′ − y′ ∈ D_{dR}(W) and the map ι : D_{dR}(W) → D_{dR}(V) is identified (via the p-adic comparison theorem, see [Kis] Thm. 3.3) with the canonical map

ι : H_{Y_{L},dR}^{2j−1}(X_{L}) → H_{dR}^{2j−1}(X_{L}).

Since Y has codimension ≥ i in X, the image of this map lands in Fil^{i} H_{dR}^{2j−1}(X_{L}), as can be seen for example by embedding L in C and using usual (mixed) Hodge theory (see [PS], §7.1 and Cor.7.4). We have thus shown

log(res_{K/L}(ξ_{γ})) ∈ Fil^{i} H_{dR}^{2j−1}(X_{L})/ Fil^{j} H_{dR}^{2j−1}(X_{L}).

But then the commutativity of the diagram

\xymatrix{H_{f}^{1}(G_{K}, V) \ar[d]^{_{res_{K/L}}} \ar[r]^{log} & H_{dR}^{2j−1}(X_{K})/ Fil^{j} H_{dR}^{2j−1}(X_{K}) \ar[d]^{can} \\ H_{f}^{1}(G_{L}, V) \ar[r]^{log} & H_{dR}^{2j−1}(X_{L})/ Fil^{j} H_{dR}^{2j−1}(X_{L})}

implies that

α_{v}(γ) = log(ξ_{γ}) ∈ (H_{dR}^{2j−1}(X_{K}) ∩ Fil^{i} H_{dR}^{2j−1}(X_{L}))/ Fil^{j} H_{dR}^{2j−1}(X_{K})
= Fil^{i} H_{dR}^{2j−1}(X_{K})/ Fil^{j} H_{dR}^{2j−1}(X_{K}).
■

3. p-adic L-functions

In this section, we summarize the main results about p-adic L-functions that will be used in the following sections. The reader may refer to [BDP2], §3A-3D for a more detailed discussion. Henceforth, we view all number fields as being contained in a fixed algebraic closure Q̄ of Q. A complex embedding Q̄⟶C and an embedding Q̄⟶C_{ℓ} for each rational prime ℓ are chosen once and for all. Let K be an imaginary quadratic field of discriminant −D, and let p be a fixed rational prime which is split in K. Write 𝔭 for the prime above p corresponding to the distinguished embedding K⟶Q̄_{p}. Also write 𝔡_{K} for the ideal (\sqrt{−D}) in 𝒪_{K}.

3.1. The Katz p-adic L-function. 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. For ν ∈ Σ(𝔠), write L_{𝔠}(ν, s) for the Hecke L-function attached to ν with the Euler factors at the primes dividing 𝔠 removed. Denote by Σ_{crit}(𝔠) the set of critical characters, i.e., those characters ν in Σ(𝔠) for which L(ν^{−1}, 0) is a critical value in the sense of Deligne. This set is the disjoint union

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

where

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

and

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

Viewing the elements of Σ_{crit}(𝔠) as p-adic characters, this set is endowed with the natural topology of uniform convergence with respect to the p-adic norm. Write Σ̂_{crit}(𝔠) for the completion of Σ_{crit}(𝔠) relative to this topology. Note that Σ_{crit}^{(1)}(𝔠) and Σ_{crit}^{(2)}(𝔠) are both dense subsets of Σ̂_{crit}(𝔠).

Let A be an elliptic curve defined over the Hilbert class field H of K such that A_{C} ≃ C/𝒪_{K}, and let ω_{A} ∈ Ω^{1}(A_{/H}) be a nowhere vanishing differential. We further assume that A has been chosen to have good reduction at the prime above p, and that ω_{A} extends to a nowhere vanishing differential on the Néron model 𝒜/𝒪_{H,p} (In the applications in this article, we shall only use the case when K has class number one, the elliptic curve A is defined over Q and ω_{A} is a global Néron differential on A. ) We fix once and for all an isomorphism 𝒜̂ ≃ 𝔾̂_{m}, where 𝒜̂ denotes the formal competion of 𝒜 along the identity section. The choice of ω_{A} along with the isomorphisms A ≃ C/𝒪_{K} and 𝒜̂ ≃ 𝔾̂_{m} determines a complex period Ω ∈ C^{×} and a p-adic period Ω_{p} ∈ C_{p}^{×} ([dS] II, 4.4), the latter being a p-adic unit.

Theorem 3.1.1. ([Katz1] §9, [dS] II 4.14) There exists a unique p-adically continuous C_{p}-valued function ℒ_{p,𝔠}(ν) on Σ̂_{crit}(𝔠) satisfying the interpolation property

(3.1.1)  \frac{ℒ_{p,𝔠}(ν)}{Ω_{p}^{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^{ℓ_{1}−ℓ_{2}}}

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

3.2. p-adic Rankin L-series. Here we summarize the properties of the p-adic Rankin L-series of [BDP1], §5. Let N be a positive integer, ε a Dirichlet character mod N and f ∈ S_{k}(Γ_{0}(N), ε) be a normalized newform of weight k ≥ 2 and character ε. Assume that the pair (f, K) satisfies the Heegner hypothesis, i.e., 𝒪_{K} contains an ideal 𝔑 such that 𝒪_{K}/𝔑 = Z/NZ. Fix once and for all such a cyclic ideal 𝔑 of norm N.

Let χ be a Hecke character of K of infinity type (ℓ_{1}, ℓ_{2}), and let ε_{χ} denote its central character. Recall that ε_{χ} is the finite order character of A_{Q}^{×} determined by the equation

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

where N is the norm character. We say that χ as above is central critical for f if

ℓ_{1} + ℓ_{2} = k  and  ε_{χ} = ε.

In this case, s = 0 is the central critical point for the complex Rankin L-series L(f, χ^{−1}, s).

Fix a rational integer c prime to pN and write Σ_{cc}(c, 𝔑, ε) for the set of central critical characters satisfying the conditions of Defn. 3.10 of [BDP2]. This set is the 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) (resp. Σ_{cc}^{(2)}(c, 𝔑, ε)) is defined to consist of those characters in Σ_{cc}(c, 𝔑, ε) of infinity type (k + j, −j) with −(k − 1) ≤ j ≤ −1 (resp. j ≥ 0.) The domain of classical interpolation for the p-adic Rankin L-series attached to f and K is the set Σ_{cc}^{(2)}(c, 𝔑, ε). For χ ∈ Σ_{cc}^{(2)}(c, 𝔑, ε), we define the algebraic part of L(f, χ^{−1}, 0) to be

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

where w(f, χ)^{−1} and C(f, χ, c) are the constants defined in equation (5.1.11) and in Theorem 4.6 of [BDP1], respectively. The set Σ_{cc}^{(2)}(c, 𝔑, ε) is dense in the p-adic completion Σ̂_{cc}(c, 𝔑, ε) of Σ_{cc}(c, 𝔑, ε).

Proposition 3.2.1. ([BDP1], Prop. 5.10) Suppose that D and c are odd integers. There is a unique C_{p}-valued, p-adically continuous function L_{p}(f, χ) on Σ̂_{cc}(c, 𝔑, ε) satisfying the interpolation property

(3.2.2)  L_{p}(f, χ) := Ω_{p}^{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 in Σ_{cc}^{(2)}(c, 𝔑, ε) of infinity type (k + j, −j).

Remark 3.2.2. The hypotheses that D and c are odd are made in loc. cit. to simplify some of the computations, and are certainly not crucial. However, they will suffice for our applications below.

3.3. Factorization of the p-adic Rankin L-series. We specialize the p-adic L-function of the previous subsection to the case that f is itself a theta series associated with a Hecke character of K; then this p-adic L-function factors as a product of two Katz p-adic L-functions.

Let ψ be a Hecke character of K of infinity type (k − 1, 0), with k ≥ 2, and consider the associated theta series

θ_{ψ} := ∑_{𝔞} ψ(𝔞)q^{N𝔞},

the sum being taken over the ideals of 𝒪_{K} prime to the conductor 𝔣_{ψ} of ψ. Assume that 𝔣_{ψ} is a cyclic ideal 𝔪 of norm M prime to D. Taking N := MD, the form f := θ_{ψ} belongs to the space of cusp forms S_{k}(Γ_{0}(N), ε), with ε = ε_{K}ε_{ψ}. Moreover, the pair (f, K) satisfies the Heegner hypothesis, the ideal 𝔑 := 𝔡_{K}𝔪 being cyclic of norm N.

Proposition 3.3.1. ([BDP2], Thm. 3.17) For all χ ∈ Σ_{cc}(c, 𝔑, ε), we have a factorization

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

If χ belongs to the range Σ_{cc}^{(2)}(c, 𝔑, ε) of classical interpolation for L_{p}(θ_{ψ}, ·), then ψ^{−1}χ belongs to Σ_{crit}^{(2)}(c𝔡_{K}) and ψ^{∗−1}χ belongs to Σ_{crit}^{(2)}(c𝔡_{K}𝔪) (see [BDP2] Lemma 3.16), so that both these characters are in the range of classical interpolation for the Katz p-adic L-function. As a consequence, the proof of the above proposition reduces to obtaining a similar factorization formula for complex special values, which is a simple consequence of the Artin formalism for these L-series. The usefulness of the proposition lies in applying the equality above to characters χ in Σ_{cc}^{(1)}(c, 𝔑, ε), which are outside the range of interpolation. In this case, the character ψ^{∗−1}χ still lies in Σ_{crit}^{(2)}(c𝔡_{K}𝔪) but ψ^{−1}χ lies in Σ_{crit}^{(1)}(c𝔡_{K}) and is outside the range of interpolation for the Katz p-adic L-function ℒ_{p,c𝔡_{K}}.

Remark 3.3.2. We observe that the critical characters ψ^{−1}χ and ψ^{∗−1}χ arising as above when χ belongs to Σ_{cc}(c, 𝔑, ε) are self-dual. (See [BDP2], Defn. 3.4 and Lemma 3.16.)

Corollary 3.3.3. Suppose that χ belongs to Σ_{cc}(c, 𝔑, ε). Then

L_{p}(θ_{ψ}, χ) ≠ 0 ⟺ ℒ_{p,𝔠}(ψ^{−1}χ) ≠ 0 and ℒ_{p,𝔠}(ψ^{∗−1}χ) ≠ 0,

where 𝔠 is any ideal in 𝒪_{K}, prime to p and divisible by c𝔡_{K}𝔪.

4. Generalized Heegner cycles and coniveau

4.1. Generalized Heegner cycles. In this section, we will give a slight variant of the construction of generalized Heegner cycles given in [BDP1]. At first we work in the more general setting of §3.2. Later, in §4.3, we specialize the discussion to the case of CM forms as in §3.3.

The setup will thus be as in §3.2, except we take c = 1 since this suffices for our applications. Let N be a positive integer, K an imaginary quadratic field (of discriminant −D) that satisfies the Heegner condition with respect to N, i.e., all the primes dividing N are either split or ramified in K and if q^{2} | N, then q is split in K. Thus there exists a cyclic ideal 𝔑 of norm N in 𝒪_{K}, i.e. 𝒪_{K}/𝔑 = Z/NZ. We fix such an ideal 𝔑 in what follows. We also fix once and for all an elliptic curve A ≃ C/𝒪_{K} defined over the Hilbert class field H of K, and with CM by 𝒪_{K}, and a generator t of the cyclic group A[𝔑]. The pair (A, t) defines a point P on the modular curve X_{1}(N), that is defined over an abelian extension of K.

For 𝔞 an ideal in 𝒪_{K}, let us write A_{𝔞} for the elliptic curve C/𝔞^{−1} and let φ_{𝔞} denote the canonical isogeny of degree N𝔞,

φ_{𝔞} : A = C/𝒪_{K} → C/𝔞^{−1} = A_{𝔞}.

Define a variety X_{r_{1},r_{2}} of dimension r_{1} + r_{2} + 1 by

X_{r_{1},r_{2}} = W_{r_{1}} × A^{r_{2}},

where W_{r_{1}} is the Kuga-Sato variety of dimension r_{1} + 1 over X_{1}(N). (We refer the reader to [BDP1], §2 and Appendix for a more explicit description of this variety and its cohomology.) The only cases that will interest us are when r_{1} ≥ r_{2} and r_{1} ≡ r_{2} mod 2. Write r_{1} + r_{2} = 2s, r_{1} − r_{2} = 2u. For each ideal 𝔞 in 𝒪_{K} that is prime to 𝔑, we construct a cycle

Δ_{r_{1},r_{2},𝔞} ∈ CH^{s+1}(X_{r_{1},r_{2}})_{0,Q}

as follows.

Let t_{𝔞} denote the image of t under the map φ_{𝔞}. Then the pair (A_{𝔞}, t_{𝔞}) defines a point on the modular curve X_{1}(N), which we denote P_{𝔞}. The fiber of W_{r_{1}} over P_{𝔞} is canonically isomorphic to A_{𝔞}^{r_{1}}. Thus viewing X_{r_{1},r_{2}} as fibered over X_{1}(N), the fiber over P_{𝔞} is

(A_{𝔞}^{r_{1}} × A^{r_{2}}) = (A_{𝔞} × A)^{r_{2}} × (A_{𝔞} × A_{𝔞})^{u}.

Now set (with tr=transpose),

Γ_{𝔞} = (graph of \sqrt{−D})^{tr} ∈ Z^{1}(A_{𝔞} × A_{𝔞}),
Γ_{φ,𝔞} = (graph of φ_{𝔞})^{tr} ∈ Z^{1}(A_{𝔞} × A),
Γ_{r_{1},r_{2},𝔞} = Γ_{φ,𝔞}^{r_{2}} × Γ_{𝔞}^{u},

and

(4.1.1)  Δ_{r_{1},r_{2},𝔞} := ϵ_{X_{r_{1},r_{2}}} (Γ_{r_{1},r_{2},𝔞}) ∈ CH^{s+1}(X_{r_{1},r_{2}})_{Q},

where ϵ_{X_{r_{1},r_{2}}} := ϵ_{W_{r_{1}}} ϵ_{A^{r_{2}}} with ϵ_{W_{r_{1}}} and ϵ_{A^{r_{2}}} being the projectors on W_{r_{1}} and A^{r_{2}} described in §2.1 and §1.4 of [BDP1] respectively. (There they are called just ϵ_{W} and ϵ_{A} respectively, but here we will need to keep track of the indices.) When r_{1} = r_{2} = r say, we just write X_{r}, Γ_{r,𝔞} and Δ_{r,𝔞} for X_{r,r}, Γ_{r,r,𝔞} and Δ_{r,r,𝔞} respectively.

The same argument as in §2.2 and §2.3 of [BDP1] shows that Δ_{r_{1},r_{2},𝔞} is homologically trivial on X_{r_{1},r_{2}}. This also follows from the following proposition which shows that the cycles constructed here are not very much more general than the generalized Heegner cycles of loc. cit., since they can be obtained by correspondence from the previously defined cycles on a higher dimensional variety.

Proposition 4.1.1. There exists an algebraic correspondence (in the ring of correspondences tensor Q)

P : X_{r_{1}} = W_{r_{1}} × A^{r_{1}} −−→ W_{r_{1}} × A^{r_{2}} = X_{r_{1},r_{2}}

such that for all ideals 𝔞 (prime to 𝔑), we have

(4.1.2)  P_{∗}(Γ_{r_{1},𝔞}) = (N𝔞)^{u}Γ_{r_{1},r_{2},𝔞},

and

(4.1.3)  P_{∗}(Δ_{r_{1},𝔞}) = (N𝔞)^{u}Δ_{r_{1},r_{2},𝔞},

as classes in CH^{s+1}(X_{r_{1},r_{2}}). In particular, since Δ_{r_{1},𝔞} is homologically trivial on X_{r_{1}}, so is Δ_{r_{1},r_{2},𝔞} on X_{r_{1},r_{2}}.

Proof. Consider the variety

Y = W_{r_{1}} × A^{s} = (W_{r_{1}} × A^{r_{2}}) × A^{u} = X_{r_{1},r_{2}} × A^{u}.

We embed Y in the product

X_{r_{1}} × X_{r_{1},r_{2}} = (X_{r_{1},r_{2}} × (A × A)^{u}) × X_{r_{1},r_{2}}

via the map

((id_{X_{r_{1},r_{2}}} , (\sqrt{−D}, id_{A})^{u}), id_{X_{r_{1},r_{2}}} ).

Then Y ∈ Z^{r_{1}+s+1}(X_{r_{1}} × X_{r_{1},r_{2}}) and it induces a correspondence P : X_{r_{1}} −−→ X_{r_{1},r_{2}}. Since p_{1}^{∗}(Δ_{r_{1}}) intersects transversally with Y in X_{r_{1}} ×X_{r_{1},r_{2}}, it is straightforward to check that the relation (4.1.2) holds using the fact that the degree of φ_{𝔞} is N𝔞. To show (4.1.3), we need to study how P interacts with the automorphisms of X_{r_{1}} that arise in the sum defining ϵ_{X_{r_{1}}} . For σ ∈ Ξ_{2u} = μ_{2}^{2u} ⋊ S_{2u}, let δ_{σ} (resp. δ̃_{σ}) be the automorphism of X_{r_{1}} = W_{r_{1}} × A^{r_{2}} × A^{2u} that acts by σ on the last 2u components of the fiber over any point of W_{r_{1}} (resp. by σ on the last 2u copies of A.) Thus we can view Ξ_{2u} × Ξ_{2u} as a subgroup of Aut(X_{r_{1}}) via (σ, τ) ↦ (δ_{σ}, δ̃_{τ}). For any σ, τ ∈ Ξ_{2u}, we have

(4.1.4)  P_{∗}(δ_{στ} δ̃_{τ} · Γ_{r_{1},𝔞}) = P_{∗}(δ_{σ} · Γ_{r_{1},𝔞}) = δ_{σ} · P_{∗}(Γ_{r_{1},𝔞}),

where we also write δ_{σ} for the automorphism of X_{r_{1},r_{2}} defined similarly, i.e. by the action of σ on the last 2u components of the fiber over W_{r_{1}}. Let G_{r_{1}} (resp. G_{r_{1},r_{2}}) be the group of automorphisms of X_{r_{1}} (resp. of X_{r_{1},r_{2}}) that occur in the sum defining ϵ_{X_{r_{1}}} (resp. ϵ_{X_{r_{1},r_{2}}} ), and let μ denote the natural homomorphisms G_{r_{1}} → μ_{2}, G_{r_{1},r_{2}} → μ_{2}. Observe that G_{r_{1},r_{2}} is naturally identified as a subgroup of G_{r_{1}}. Then, viewing Ξ_{2u} as a subgroup of Aut(X_{r_{1},r_{2}}) via σ ↦ δ_{σ}, we have

P_{∗}ϵ_{X_{r_{1}}} Γ_{r_{1},𝔞} = P_{∗} (\frac{1}{|G_{r_{1}}|} ∑_{g∈G_{r_{1}}} μ(g)g · Γ_{r_{1},𝔞})

= P_{∗} (\frac{1}{|G_{r_{1}}/(Ξ_{2u} × Ξ_{2u})|} ∑_{g∈G_{r_{1}} /(Ξ_{2u}×Ξ_{2u})} μ(g)g · \frac{1}{|Ξ_{2u}|^{2}} ∑_{(σ,τ)∈Ξ_{2u}} μ(δ_{σ}δ̃_{τ})δ_{σ}δ̃_{τ} · Γ_{r_{1},𝔞})

= \frac{1}{|G_{r_{1},r_{2}}/Ξ_{2u}|} ∑_{g∈G_{r_{1},r_{2}} /Ξ_{2u}} μ(g)g · P_{∗} (\frac{1}{|Ξ_{2u}|^{2}} ∑_{(σ,τ)∈Ξ_{2u}} μ(δ_{στ} δ̃_{τ})δ_{στ} δ̃_{τ} · Γ_{r_{1},𝔞})

= \frac{1}{|G_{r_{1},r_{2}}/Ξ_{2u}|} ∑_{g∈G_{r_{1},r_{2}} /Ξ_{2u}} μ(g)g · (\frac{1}{|Ξ_{2u}|} ∑_{σ∈Ξ_{2u}} μ(δ_{σ})δ_{σ} · P_{∗}(Γ_{r_{1},𝔞}))  (by (4.1.4))

= \frac{1}{|G_{r_{1},r_{2}}|} ∑_{g∈G_{r_{1},r_{2}}} μ(g)g · P_{∗}(Γ_{r_{1},𝔞}) = (N𝔞)^{u}ϵ_{X_{r_{1},r_{2}}} Γ_{r_{1},r_{2},𝔞},  (by (4.1.2))

which proves (4.1.3). ■

We now assume that A has good reduction at the prime above p and pick a generator ω_{A} for Ω^{1}(A_{/H}) as described in §3.1. This determines a period pair (Ω, Ω_{p}) ∈ C^{×} × 𝒪_{C_{p}}^{×} as well as a unique element η_{A} ∈ H_{dR}^{1}(A_{/H}) such that

[α]^{∗}η_{A} = α^{ρ}η_{A} and ⟨ω_{A}, η_{A}⟩ = 1,

where ρ denotes the nontrivial element of Gal(K/Q). For any power A^{r} and for any j such that 0 ≤ j ≤ r, we define ω_{A}^{j}η_{A}^{r−j} by

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

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

Proposition 4.1.2. Let F be a number field over which all the structures above can be defined and v a place of F at which X_{r_{1}} and X_{r_{1},r_{2}} have good reduction. Then for any ω ∈ H_{dR}^{r_{1}+1}(W_{r_{1},F}), we have

(4.1.5)  (N𝔞)^{u} · β_{v}(Δ_{r_{1},r_{2},𝔞})[ω ∧ ω_{A}^{j}η_{A}^{r_{2}−j}] = (2\sqrt{−D})^{u} · β_{v}(Δ_{r_{1},𝔞})[ω ∧ ω_{A}^{j+u}η_{A}^{r_{2}−j+u}],

where β_{v} is the map defined in §2.4.

Proof. We first remark that since ϵ_{X_{r_{1},r_{2}}} (resp. ϵ_{X_{r}}) is the product of the commuting idempotents ϵ_{W_{r_{1}}} and ϵ_{A^{r_{2}}} (resp. ϵ_{A^{r_{1}}} ), we have

(4.1.6)  ϵ_{A^{r_{2}}} · Δ_{r_{1},r_{2},𝔞} = Δ_{r_{1},r_{2},𝔞}  and  ϵ_{A^{r_{1}}} · Δ_{r_{1},𝔞} = Δ_{r_{1},𝔞}.

Write ϖ for the class p_{1}^{∗}ω_{A} ∧ ⋯ p_{j}^{∗}ω_{A} ∧ p_{j+1}^{∗}η_{A} ∧ ⋯ ∧ p_{r_{2}}^{∗}η_{A} ∈ H^{r_{2}}(A^{r_{2}}). Then using (4.1.6) twice as well as Prop. 4.1.1, we have

(4.1.7)  (N𝔞)^{u} · β_{v}(Δ_{r_{1},r_{2},𝔞})[ω ∧ ω_{A}^{j}η_{A}^{r_{2}−j}] = (N𝔞)^{u} · β_{v}(Δ_{r_{1},r_{2},𝔞})[ω ∧ ϵ_{A^{r_{2}}}^{∗} (ϖ)]
= (N𝔞)^{u} · β_{v}(Δ_{r_{1},r_{2},𝔞})[ω ∧ ϖ]
= β_{v}(P_{∗}(Δ_{r_{1},𝔞}))[ω ∧ ϖ]
= β_{v}(Δ_{r_{1},𝔞}) [P^{∗} (ω ∧ ϖ)]
= β_{v}(Δ_{r_{1},𝔞}) [ϵ_{A^{r_{1}}}^{∗} P^{∗}(ω ∧ ϖ)]

Let ϑ denote the cohomology class of the transpose of the graph of \sqrt{−D} on A × A. Then

P^{∗}(ω ∧ ϖ) = ω ∧ ϖ ∧ (p_{r_{2}+1}, p_{r_{2}+2})^{∗}ϑ ∧ ⋯ ∧ (p_{r_{1}−1}, p_{r_{1}})^{∗}ϑ.

But the class ϑ on A × A is of the form

ϑ = ϑ_{1} + ϑ_{2},

where

ϑ_{1} = (\sqrt{−D}) · (p_{1}^{∗}(ω_{A}) ∧ p_{2}^{∗}(η_{A}) + p_{1}^{∗}(η_{A}) ∧ p_{2}^{∗}(ω_{A})) ,

and ϑ_{2} is in the span of p_{1}^{∗}(ω_{A}) ∧ p_{1}^{∗}(η_{A}) and p_{2}^{∗}(ω_{A}) ∧ p_{2}^{∗}(η_{A}). Then

ϵ_{A^{r_{1}}}^{∗} P^{∗}(ω ∧ ϖ) = ϵ_{A^{r_{1}}}^{∗} [ω ∧ ϖ ∧ (p_{r_{2}+1}, p_{r_{2}+2})^{∗}ϑ ∧ ⋯ ∧ (p_{r_{1}−1}, p_{r_{1}})^{∗}ϑ]
= ϵ_{A^{r_{1}}}^{∗} [ω ∧ ϖ ∧ (p_{r_{2}+1}, p_{r_{2}+2})^{∗}(ϑ_{1} + ϑ_{2}) ∧ ⋯ ∧ (p_{r_{1}−1}, p_{r_{1}})^{∗}(ϑ_{1} + ϑ_{2})]
= ϵ_{A^{r_{1}}}^{∗} [ω ∧ ϖ ∧ (p_{r_{2}+1}, p_{r_{2}+2})^{∗}ϑ_{1} ∧ ⋯ ∧ (p_{r_{1}−1}, p_{r_{1}})^{∗}ϑ_{1}]
= (2\sqrt{−D})^{u}ϵ_{A^{r_{1}}}^{∗} [ω ∧ ϖ ∧ p_{r_{2}+1}^{∗}ω_{A} ∧ p_{r_{2}+2}^{∗}η_{A} ∧ ⋯ ∧ p_{r_{1}−1}^{∗}ω_{A} ∧ p_{r_{1}}^{∗}η_{A}]
= (2\sqrt{−D})^{u} · ω ∧ ω^{j+u}η^{r_{2}−j+u}.

The proposition follows by combining this last equality with (4.1.7). ■

Theorem 4.1.3. Let f ∈ S_{r_{1}+2}(Γ_{0}(N), ε_{f}) be a newform of level N, and let χ be a Hecke character of K of infinity type (r_{2} + 1 − j, 1 + j) for some j with 0 ≤ j ≤ r_{2} and of (finite) type (1, 𝔑, ε_{f}) ([BDP1] Defn. 4.4). Let p = 𝔭𝔭̄ be a prime split in K and not dividing N, and v a place of F lying above 𝔭. Then

(∑_{𝔞} χ^{−1}(𝔞)N𝔞 · β_{v}(Δ_{r_{1},r_{2},𝔞})[ω_{f} ∧ ω_{A}^{j}η_{A}^{r_{2}−j}])^{2} = (−4D)^{u}ℰ_{p}(f, χN_{K}^{u})^{−1} \frac{L_{p}(f, χN_{K}^{u})}{Ω_{p}^{r_{2}−2j}},

where the sum is over a set of representatives {𝔞} for the ideal classes in 𝒪_{K} chosen to be prime to 𝔑, and the Euler-type factor ℰ_{p}(f, χN_{K}^{u}) is given by

ℰ_{p}(f, χN_{K}^{u}) = (1 − χ^{−1}(𝔭̄)p^{−u}a_{p}(f) + χ^{−2}(𝔭̄)ε_{f}(p)p^{r_{2}+1})^{2}.

Proof. Let χ′ denote the character χN_{K}^{u}. Then χ′ is also of finite type (1, 𝔑, ε_{f}) but has infinity type

(r_{2} + 1 − j + u, 1 + j + u) = (r_{1} + 1 − (j + u), 1 + (j + u)),

so that χ′ lies in Σ_{cc}^{(1)}(1, 𝔑, ε_{f}) in the notation of §3.2. By the previous proposition, we have

(χ^{−1}(𝔞)N𝔞) · β_{v}(Δ_{r_{1},r_{2},𝔞})[ω_{f} ∧ ω_{A}^{j}η_{A}^{r_{2}−j}] = (2\sqrt{−D})^{u} · (χ′)^{−1}(𝔞)N𝔞 · β_{v}(Δ_{r_{1},𝔞})[ω_{f} ∧ ω_{A}^{j+u}η_{A}^{r_{1}−(j+u)}].

The result now follows by the main theorem (Thm. 5.13) of [BDP1] applied to the pair (f, χ′). ■

Remark 4.1.4. When u = 0, so that r_{1} = r_{2} = r say, one recovers exactly the statement of Th. 5.13 of loc. cit. for the case of conductor c = 1.

4.2. Fields of definition. We now systematically deal with questions of rationality for the structures appearing in the previous section:

(i) The variety X_{r_{1},r_{2}} is defined over Q.

(ii) The elliptic curve A (along with its complex multiplication) can be defined over the Hilbert class

field H of K. (iii) The point t is defined over an abelian extension of H that may not be abelian over K. However,

the pair (A, t) (up to isomorphism, equivalently the point P_{A} on X_{1}(N)) is defined over the abelian extension F/K corresponding to the subgroup

K^{×}W ⊆ A_{K}^{×},

where

W := {x ∈ A_{K}^{×} : x𝒪_{K} = 𝒪_{K}, xt = t}.

Let W_{0} be the subgroup

W_{0} := {x ∈ A_{K}^{×} : x𝒪_{K} = 𝒪_{K}}

so that H is the class field corresponding to K^{×}W_{0}. Then we have canonical isomorphisms

(4.2.1)  Gal(F/H) ≃ K^{×}W_{0}/K^{×}W ≃ W_{0}/(W_{0} ∩ K^{×}W)
≃ (𝒪_{K}/𝔑𝒪_{K})^{×}/{±1} ≃ (Z/NZ)^{×}/{±1}.

(iv) The elliptic curve A_{𝔞} can also be defined over H. The pair (A_{𝔞}, t_{𝔞}) is defined over F, since the

subgroup

{x ∈ A_{K}^{×} : x𝔞 = 𝔞, xt_{𝔞} = t_{𝔞}}

is equal to W for any 𝔞 prime to 𝔑.

(v) The isogeny φ_{𝔞} is defined over H, while the correspondences ϵ_{X_{r_{1}}} and ϵ_{A^{r_{2}}} are defined over Q. Thus the cycle Δ_{r_{1},r_{2},𝔞} is defined over F.

(vi) The character ε_{f} is a character of (Z/NZ)^{×}. Let ker(ε_{f}) denote its kernel and let H_{f} be the intermediate extension H ⊆ H_{f} ⊆ F that corresponds to the subgroup

(ker ε_{f}){±1}/{±1} ⊆ (Z/NZ)^{×}/{±1},

via the isomorphism (4.2.1) above. Define

(4.2.2)  Δ̃_{r_{1},r_{2},𝔞} := \frac{1}{[F : H_{f}]} Tr_{F/H_{f}} (Δ_{r_{1},r_{2},𝔞}).

Thus Δ̃_{r_{1},r_{2},𝔞} is defined over the field H_{f}. (vii) Let Q(f) denote the extension of Q generated by the Hecke eigenvalues of f, so that ω_{f} is defined

over Q(f). Finally, set F′ := F · Q(f). Let v be a place of F′ lying over the prime 𝔭.

Proposition 4.2.1. For σ ∈ Gal(F/H_{f}), the value

(4.2.3)  β_{v}(Δ_{r_{1},r_{2},𝔞}^{σ})[ω_{f} ∧ ω^{j}η^{r_{2}−j}]

is independent of the choice of σ.

Proof. By Prop. 4.1.2 and the fact that the correspondence P is defined over H, it suffices to show this in the special case r_{1} = r_{2} = r, say. Suppose that σ ∈ Gal(F/H) is the image of the element a ∈ (Z/NZ)^{×}. Then σ sends Δ_{r_{1},r_{2},𝔞} to a cycle defined similarly, but with t (resp. t_{𝔞}) replaced by at (resp. at_{𝔞}). It follows from Lemma 3.22 and Prop. 3.24 of [BDP1] that

β_{v}(Δ_{r_{1},r_{2},𝔞}^{σ})[ω_{f} ∧ ω^{j}η^{r_{2}−j}] = ε_{f}(a) · β_{v}(Δ_{r_{1},r_{2},𝔞})[ω_{f} ∧ ω^{j}η^{r_{2}−j}].

For σ ∈ Gal(F/H_{f}), we can choose a to lie in ker(ε_{f}), so the expression (4.2.3) is indeed independent of the choice of σ. ■

Corollary 4.2.2.

β_{v}(Δ̃_{r_{1},r_{2},𝔞})[ω_{f} ∧ ω^{j}η^{r_{2}−j}] = β_{v}(Δ_{r_{1},r_{2},𝔞})[ω_{f} ∧ ω^{j}η^{r_{2}−j}].

We now pick a set S of coset representatives {𝔞} for Pic(𝒪_{K}) that are all prime to 𝔑 to make the following definition.

Definition 4.2.3. Let χ be a character of infinity type (r_{2} + 1 − j, 1 + j) and of finite type (𝔑, ε_{f}). Then set

Δ̃_{r_{1},r_{2}}^{χ} := ∑_{𝔞∈S} χ^{−1}(𝔞)N𝔞 · Δ̃_{r_{1},r_{2},𝔞} ∈ CH^{s+1}(X_{r_{1},r_{2}}_{/H_{f}})_{0} ⊗ Q(χ),

where Q(χ) is the (number) field generated over Q by the values of χ on S.

Remark 4.2.4. Note that the cycle Δ̃_{r_{1},r_{2}}^{χ} does depend on the choice of S. However, as follows from the discussion in Sec. 5.1 of [BDP1], the value β_{v}(Δ̃_{r_{1},r_{2}}^{χ})[ω_{f} ∧ ω_{A}^{j}η_{A}^{r_{2}−j}] is independent of this choice. This independence is a reflection of the fact that the “f” part of χ^{−1}(𝔞)Δ_{r_{1},r_{2},𝔞}, i.e., the image of χ^{−1}(𝔞)Δ_{r_{1},r_{2},𝔞} under the projector defining the Grothendieck motive M_{f} (constructed in [Scholl]), is independent of the choice of S, at least up to a cycle in the kernel of the Abel-Jacobi map. The last statement is an easy exercise if one recalls that χ is of finite type (𝔑, ε_{f}) and uses the following: viewing X_{r_{1},r_{2}} as fibered over X_{1}(N), the Abel-Jacobi image of a cycle supported on a fiber above a point P_{𝔞} and in the image of the projector ϵ_{X_{r_{1},r_{2}}} only depends on the cohomology class of the cycle in this fiber.

Theorem 4.2.5. For v any place of H_{f} · Q(f) lying above 𝔭,

(β_{v}(Δ̃_{r_{1},r_{2}}^{χ})[ω_{f} ∧ ω_{A}^{j}η_{A}^{r_{2}−j}])^{2} = (−4D)^{u}ℰ_{p}(f, χN_{K}^{u})^{−1} \frac{L_{p}(f, χN_{K}^{u})}{Ω_{p}^{r_{2}−2j}}.

Proof. This follows immediately from Thm. 4.1.3 and Cor. 4.2.2. ■

4.3. Cycles associated with Grossencharacters. We now return to the situation of the introduction, namely we assume that K has odd discriminant −D, class number one, and that w_{K} = 2, and specialize the constructions from the previous sections to this case. Recall that 𝔡_{K} = (\sqrt{−D)} and ε_{K} is the quadratic Dirichlet character corresponding to the extension K/Q. There is then a canonical Hecke character ψ of K of infinity type (1, 0) and conductor 𝔡_{K}, whose square ψ^{2} is the unique unramified Hecke character of K of infinity type (2, 0). To construct this character explicitly, we note that for all such K, the composite map Z → 𝒪_{K} → 𝒪_{K}/𝔡_{K} induces an isomorphism

(4.3.1)  (Z/DZ)^{×} ≃ (𝒪_{K}/𝔡_{K})^{×}.

Then

ψ((a)) = ε_{K}(a mod 𝔡_{K}) · a

for any ideal (a), with a ∈ 𝒪_{K} prime to 𝔡_{K}. Here we view a mod 𝔡_{K} as an element in (Z/DZ)^{×} via (4.3.1).

The motive we are interested in is that attached to ψ^{2r+1} for some positive integer r. Let i be any integer satisfying

0 ≤ i ≤ r and i ≡ r − 1 mod 2.

Let f be the modular form

f = θ_{ψ^{r+i+1}} = ∑_{𝔞} ψ^{r+i+1}(𝔞)e^{2πiN𝔞z} ∈ S_{r+i+2}(Γ_{0}(D), ε_{K}).

Thus N = D, ε_{f} = ε_{K}, Q(f) = K and

ker(ε_{f}){±1}/{±1} = (Z/DZ)^{×}/{±1},

so that H_{f} = H = K. Let χ the character ψ^{∗}^{r−i}N_{K}. Picking 𝔑 = 𝔡_{K}, one sees that χ is of finite type (1, 𝔑, ε_{f}).

Consider the variety

X_{r+i,r−i} = W_{r+i} × A^{r−i}

of dimension 2r + 1, where A is the unique elliptic curve over Q with CM by 𝒪_{K} and minimal conductor D^{2}. Note that X_{r+i,r−i} is in fact defined over Q. Since K has class number one, we may pick S = {1} and we have

(4.3.2)  Δ̃_{r+i,r−i}^{χ} = Δ̃_{r+i,r−i,𝒪_{K}} ∈ CH^{r+1}(X_{r+i,r−i}_{/K})_{0} ⊗ Q.

For ease of notation, we will simply call this cycle Δ̃_{r+i,r−i}.

Since X_{r+i,r−i} is defined over Q, there is an action of complex conjugation (denoted τ) on the Chow group in (4.3.2), and we consider its effect on the cycle Δ̃_{r+i,r−i}. For any generator t′ of A[𝔡_{K}], τ sends the point (A, t′) on X_{1}(D) to (A, t′′) where t′′ is some other generator of A[𝔡_{K}]. Since τ preserves the graph of the identity on A × A but sends the graph of \sqrt{−D} on A × A to the graph of −\sqrt{−D}, we see that Δ̃_{r+i,r−i} is in the + or − eigen-space for the action of τ depending on whether i is even or odd. However, there is an obvious action of End(A) on CH^{r+1}(X_{r+i,r−i}), for instance via its action on any one of the elliptic curve factors of X_{r+i,r−i} and the induced pullback or pushforward on cycles. (Note that there is at least one such factor since i ≡ r − 1 mod 2.) Defining

Δ̃′_{r+i,r−i} := [\sqrt{−D}]^{∗} · Δ̃_{r+i,r−i},

we see that Δ̃′_{r+i,r−i} is in the opposite eigenspace for the action of τ as Δ̃_{r+i,r−i}.

Let p = 𝔭𝔭̄ be a prime split in K, and let ℒ_{p,𝔡_{K}} denote the Katz p-adic L-function as in Sec. 3. We supress 𝔡_{K} from the notation and simply write ℒ_{p} henceforth.

Theorem 4.3.1. Let i be an integer in the range 0 ≤ i ≤ r and satisfying i ≡ r − 1 mod 2. Suppose that

L(ψ^{2i+1}, i + 1) ≠ 0  and  ℒ_{p}(ψ^{−r}ψ^{∗}^{r+1}) ≠ 0.

Then the cycles Δ̃_{r+i,r−i}, Δ̃′_{r+i,r−i} ∈ CH^{r+1}(X_{r+i,r−i}_{/K})_{0} ⊗ Q are nontrivial in the top graded piece of the coniveau filtration, i.e. the classes

[Δ̃_{r+i,r−i}], [Δ̃′_{r+i,r−i}] ∈ N^{0} CH^{r+1}(X_{r+i,r−i})_{Q}/N^{1} CH^{r+1}(X_{r+i,r−i})_{Q}

are nonzero. In particular, they are nonzero in Gr^{r+1}(X_{r+i,r−i}) ⊗ Q.

Proof. It clearly suffices to prove this for one of the cycles, say Δ̃_{r+i,r−i}. We compute β_{𝔭}(Δ̃_{r+i,r−i})[ω_{f} ∧ ω_{A}^{r−i}]. By Theorem 4.2.5,

(β_{𝔭}(Δ̃_{r+i,r−i})[ω_{f} ∧ ω_{A}^{r−i}])^{2} ∼ L_{p}(f, χN_{K}^{i}),

where we use ∼ (in this proof) to denote equality up to multiplication by a nonzero factor. But by Prop. 3.3.1,

L_{p}(f, χN_{K}^{i}) ∼ ℒ_{p}(ψ^{−(r+i+1)}χN_{K}^{i})ℒ_{p}(ψ^{∗}^{−(r+i+1)}χN_{K}^{i})
= ℒ_{p}(ψ^{−(r+i+1)}ψ^{∗}^{r−i}N_{K}^{i+1})ℒ_{p}(ψ^{∗}^{−(r+i+1)}ψ^{∗}^{r−i}N_{K}^{i+1})
= ℒ_{p}(ψ^{−r}ψ^{∗}^{r+1})ℒ_{p}(ψ^{i+1}ψ^{∗}^{−i}).

Now in the second term above the character ψ^{i+1}ψ^{∗}^{−i} is in the range of interpolation for ℒ_{p}, and so up to a nonzero factor, it equals (the algebraic part of) the L-value

L(ψ^{−(i+1)}ψ^{∗}^{i}, 0) = L(ψ^{∗}^{2i+1}, i + 1) = L(ψ^{2i+1}, i + 1).

The result now follows by using Cor. 2.4.2. ■ Let Ω(A) denote the real period of the elliptic curve A. This is related to the period Ω by

Ω = Ω(A)/\sqrt{−D}.

Let us define the algebraic part of the L-value L(ψ^{2k+1}, k + 1) by

(4.3.3)  L^{alg}(ψ^{2k+1}, k + 1) = 2 · \frac{k!(2π\sqrt{D})^{k}}{Ω(A)^{2k+1}} · L(ψ^{2k+1}, k + 1).

Proposition 4.3.2. The algebraic part L^{alg}(ψ^{2k+1}, k+1) lies in Q, and has denominator (multiplicatively) bounded above by 2^{k−1}D^{2k+1}.

Proof. That L^{alg} lies in Q follows from [GS], Thm. 8.1. That the denominator is bounded by 2^{k−1}D^{2k+1} follows from loc. cit., Prop. 10.11 (a). ■

In particular, the values of L^{alg} are p-adic integers for all primes p not dividing 2D. In practice, we find that the values of L^{alg} are in fact integers and also perfect squares, as is seen in the tables in Appendix A. The values in those tables were computed using MAGMA. In order to provably identify these values, one may use the following rapidly converging series: (see [Gre], equation (1), for example)

(4.3.4)  L(ψ^{2k+1}, k + 1) = ∑_{𝔞} (ψ^{2k+1}(𝔞) + ψ^{2k+1}(𝔞̄)) N(𝔞)^{−(k+1)}e^{−2πN(𝔞)/D^{2}} ∑_{j=0}^{k} \frac{(2πN(𝔞)/D^{2})^{j}}{j!} ,

where the sum is over integral ideals 𝔞 of 𝒪_{K}. By estimating the tail of the series (4.3.4) and using Prop. 4.3.2 above which bounds the denominator of the L-value, we can provably identify the L-value exactly.

Corollary 4.3.3. Let i be an integer in the range 0 ≤ i ≤ r and satisfying i ≡ r − 1 mod 2. Suppose that there are odd primes p_{1}, p_{2} (not necessarily distinct) split in K, and positive integers k_{1} and k_{2} satisfying

k_{1} ≡ i mod (p_{1} − 1), k_{2} ≡ −r − 1 mod (p_{2} − 1)

such that

(4.3.5)  L^{alg}(ψ^{2k_{1}+1}, k_{1} + 1) ≠ 0 mod p_{1},  L^{alg}(ψ^{2k_{2}+1}, k_{2} + 1) ≠ 0 mod p_{2}.

Then the conclusion of Theorem 4.3.1 holds for the pair (r, i).

Proof. Let (p, k) be the pair (p_{1}, k_{1}) or (p_{2}, k_{2}). By the interpolation property of ℒ_{p}, we have

\frac{ℒ_{p}(ψ^{k+1}ψ^{∗}^{−k})}{Ω_{p}^{2k+1}} = \frac{k!(2π/\sqrt{−D})^{k}}{Ω^{2k+1}} (1 − ψ^{k+1}ψ^{∗}^{−k}(𝔭)/p)(1 − ψ^{−(k+1)}ψ^{∗}^{k}(𝔭̄))L(ψ^{−(k+1)}ψ^{∗}^{k}, 0)
∼ (1 − ψ^{k+1}ψ^{∗}^{−k}(𝔭)/p)(1 − ψ^{−(k+1)}ψ^{∗}^{k}(𝔭̄))L^{alg}(ψ^{2k+1}, k + 1),

where we use ∼ to denote that the LHS = (a 𝔭-adic unit) × RHS. The Euler and Euler-like factor are both seen to be 𝔭-adic units, since ψ(𝔭)/p and ψ^{∗}(𝔭) are both 𝔭-adic units. Since Ω_{p} is a unit in 𝒪_{C_{p}}, we conclude that ℒ_{p}(ψ^{k+1}ψ^{∗}^{−k}) is a unit in 𝒪_{C_{p}}. The congruence

ℒ_{p}(ψ^{k+1}(ψ^{∗})^{−k}) ≡ ℒ_{p}(ψ^{k+1}(ψ^{∗})^{−k} · (ψ/ψ^{∗})^{(p−1)ℓ}) mod 𝔪_{𝒪_{Cp}}

implies that

ℒ_{p_{1}}(ψ^{i+1}ψ^{∗}^{−i}) ≢ 0 mod 𝔪_{𝒪_{Cp_{1}}}  and  ℒ_{p_{2}}(ψ^{−r}ψ^{∗}^{r+1}) ≢ 0 mod 𝔪_{𝒪_{Cp_{2}}} .

Thus, certainly

ℒ_{p_{2}}(ψ^{−r}ψ^{∗}^{r+1}) ≠ 0,

and by the interpolation property for ℒ_{p_{1}} we see that

L(ψ^{2i+1}, i + 1) ≠ 0.

The conclusion then follows by applying Theorem 4.3.1 with p = p_{2}. ■

Corollary 4.3.4. Let ε = 0 if sign L(ψ, s) = +1 and ε = 1 if sign L(ψ, s) = −1. Suppose that there is an odd prime p split in K, such that

(4.3.6)  L^{alg}(ψ^{2k+1}, k + 1) ≠ 0 mod p

for

k ∈ S_{ε,p} := {ε, ε + 2, …, ε + (p − 3)}.

Let r be a positive integer such that

r ≡ ε + 1 mod 2,

and i an integer in the range 0 ≤ i ≤ r such that

i ≡ ε mod 2.

Then the conclusion of Theorem 4.3.1 holds for the pair (r, i).

Proof. Since i ≡ ε mod 2 and −r − 1 ≡ ε mod 2, there exist k_{1}, k_{2} ∈ S_{ε,p} such that

k_{1} ≡ i mod (p − 1)  and  k_{2} ≡ −r − 1 mod (p − 1).

Now apply the previous Corollary with p_{1} = p_{2} = p. ■ We now define Δ̃_{r+i,r−i}^{±}. Set

Δ̃_{r+i,r−i}^{+} = \{ \begin{matrix}Δ̃_{r+i,r−i},  if i is even; \\ Δ̃′_{r+i,r−i},  if i is odd. \end{matrix}

Likewise, set

Δ̃_{r+i,r−i}^{−} = \{ \begin{matrix}Δ̃_{r+i,r−i},  if i is odd; \\ Δ̃′_{r+i,r−i},  if i is even. \end{matrix}

Corollary 4.3.5. Suppose that r is odd if D = 7 and that r is even if D = 11, 19, 43, 67, 163. Then for any positive integer i such that

0 ≤ i ≤ r  and  i ≡ r − 1 mod 2,

the cycles Δ̃_{r+i,r−i}^{+} and Δ̃_{r+i,r−i}^{−} are defined over K, are in the respective ±1-eigenspaces for the action of complex conjugation, and are nonzero in the top graded piece of the coniveau filtration; i.e.,

[Δ_{r+i,r−i}^{+}], [Δ_{r+i,r−i}^{−}] ∈ N^{0} CH^{r+1}(X_{r+i,r−i})_{Q}/N^{1} CH^{r+1}(X_{r+i,r−i})_{Q}

are nonzero. In particular, these cycles are nonzero in Griffiths group Gr^{r+1}(X_{r+i,r−i}) ⊗ Q.

Proof. We verify that for each value of D, there is some odd prime p split in K such that the condition (4.3.6) is verified for all k in S_{ε,p}. The L-values in question were computed as explained earlier in this section and are listed in the tables in Appendix A. (Rather, what is listed is the prime factorization of the square-root of L^{alg}.) Indeed, we find the required nondivisibility holds for D = 7 with p = 11, for D = 11, 19 with p = 5, for D = 43 with p = 13, for D = 67 with p = 23 and for D = 163 with p = 41! ■

5. Miscellany

5.1. A non-CM example. We consider a special case of the more general construction of §4.1. Take N = 5 and let K = Q(\sqrt{−11}). Note that the prime 5 is split in K, so the constructions of §4.1 apply. Consider the Kuga-Sato variety W_{2} over X_{1}(5), which has dimension 3 and is defined over Q. The middle dimensional cohomology H^{3}(W_{2}) is one-dimensional, corresponding to the unique cusp form f on Γ_{1}(5) of weight 4. The form f in fact has trivial central character, is not of CM type and has q-expansion:

f(q) = q − 4q^{2} + 2q^{3} + 8q^{4} − 5q^{5} − 8q^{6} + 6q^{7} − 23q^{9} + 20q^{10} + 32q^{11} + …

Theorem 5.1.1.

rank Gr^{2}(W_{2}_{/K}) ≥ 1.

Recall that X_{2,2} = W_{2} × A^{2}, where A is an elliptic curve over Q with CM by 𝒪_{K} and (minimal) conductor 121. We will simultaneously show the following result as well.

Theorem 5.1.2.

dim CH^{3}(X_{2,2})_{0,Q}/N^{1} CH^{3}(X_{2,2})_{Q} ≥ 1.

In particular,

rank Gr^{3}(X_{2,2}) ≥ 1.

For the similar groups of cycles defined over K, the ranks of the ±1-eigenspaces for the action of complex conjugation are each ≥ 1.

Remark 5.1.3. The L-function L(f, s) has sign +1 and is in fact nonzero at the center. Thus one does not expect to be able to construct a nontorsion cycle in Gr^{2}(W_{2}) that is defined over Q. However, the L-function L_{K}(f, s) has sign −1 since K satisfies the Heegner hypothesis for f and so one does expect to get such a nontorsion cycle defined over K, as given by Thm. 5.1.1. On the other hand, for the variety X_{2,2}, the relevant cohomology is H^{5}, which contains the motive M_{f} ⊗ M_{θ_{ψ^{2}}} , with ψ the Hecke character of K corresponding to A. The corresponding L-function

L(s, M_{f} ⊗ M_{θ_{ψ^{2}}} ) = L(f ⊗ θ_{ψ^{2}}, s)

has sign −1, again due to the Heegner condition, so that one does expect to construct a nontorsion cycle in Gr^{3} that is defined over Q as well as one that is defined over K and is in the −1-eigenspace for complex conjugation, as is given by Thm. 5.1.2.

Proof. (of Thms. 5.1.1 and 5.1.2) Let χ_{1} and χ_{2} be the Hecke characters of K defined by

χ_{1} := N_{K}, χ_{2} := ψ^{2} · N_{K}.

Both χ_{1} and χ_{2} are unramified characters, and are certainly of type (1, 𝔑, ε_{f}) for any of the two choices of 𝔑. Fix an 𝔑 and consider the cycles

Δ̃_{2,0}^{χ_{1}} ∈ CH^{2}(W_{2}_{/K})_{0,Q},  and  Δ̃_{2,2}^{χ_{2}} ∈ CH^{3}(X_{2,2}_{/K})_{0,Q}.

We will show that these cycles are nontrivial in N^{0}/N^{1}. We first give the argument for Δ̃_{2,0}^{χ_{1}} and then ouline the changes necessary to deal with Δ̃_{2,2}^{χ_{2}}. By Cor. 2.4.2 and Thm. 4.2.5, to show that Δ̃_{2,0}^{χ_{1}} is nontrivial in N^{0}/N^{1}, it suffices to show that

(5.1.1)  L_{p}(f, χ_{1}) ≠ 0

for some prime p split in K. In fact, by [BDP1] (Thm. 5.9 and Prop. 5.10), to show (5.1.1) is equivalent to showing that

(5.1.2)  θ^{−2}f^{♭}(A, A[𝔑], ω_{A}) ≠ 0

where θ = q \frac{d}{dq} is the Atkin-Serre operator and f^{♭} is the modular form defined by

(5.1.3)  f^{♭} := ∑_{(n,p)=1} a_{n}(f)q^{n}.

Recall that for any integer k, θ^{k}f^{♭} is a p-adic modular form of weight 4 + 2k and satisfies the p-adic continuity property

(5.1.4)  θ^{k_{1}}f^{♭}(A, A[𝔑], ω_{A}) ≡ θ^{k_{2}}f^{♭}(A, A[𝔑], ω_{A}) mod 𝔭^{n},  if k_{1} ≡ k_{2} mod (p − 1)p^{n−1}.

Thus (5.1.2) would follow if one could show that for some choice of prime p split in K and some choice of positive integer k, one has

k ≡ −2 mod (p − 1)  and  θ^{k}f^{♭}(A, A[𝔑], ω_{A}) ≠ 0 mod 𝔭.

Now, for positive k, we have a congruence

(5.1.5)  θ^{k}f^{♭}(A, A[𝔑], ω_{A}) ≡ θ^{k}f(A, A[𝔑], ω_{A}) mod 𝔭,

since (by the proof of [BDP1], Theorem 5.9) these two values differ by an Euler-type factor which is seen to equal 1 modulo 𝔭. In addition, by [Katz2], Theorem 2.6.36, we have

(5.1.6)  θ^{k}f(A, A[𝔑], ω_{A}) = δ^{k}f(A, A[𝔑], ω_{A}),

where

(5.1.7)  δ^{k}f := δ_{4+2k−2} ∘ ⋯ ∘ δ_{4}f,  δ_{j} := \frac{1}{2πi} (\frac{d}{dz} + \frac{j}{2iy} ) .

The operator δ_{j} is the Shimura-Maass operator that takes nearly holomorphic modular forms of weight j to nearly holomorphic modular forms of weight j + 2.

We now take k = 0 and p = 3; note that 3 is split in K. The value δ^{k}f(A, A[𝔑], ω_{A}) = f(A, A[𝔑], ω_{A}) lies in K and is certainly integral outside of 5. Since its denominator at 5 can be explicitly bounded by [DR] (Ch. 7, Thm. 3.9 and §3.17,3.18), one can provably identify this value by a finite computation. We pick 𝔑 = (α) where α := \frac{3−\sqrt{−11}}{2}, so that τ := (−3 + \sqrt{−11})/10 is a representative on Γ_{0}(5)\ℌ for the associated Heegner point (C/𝔑^{−1} → C/𝒪_{K}). Then

Zτ + Z = \frac{1}{5}𝔑,

while the period lattice of (A, ω_{A}) is \frac{1}{\sqrt{−11}}𝒪_{K} · Ω_{A}, where Ω_{A} is the real period of ω_{A}. Then one finds that

(5.1.8)  f(A, A[𝔑], ω_{A}) = (\frac{2πi}{Ω_{0}})^{4} f(τ),  where Ω_{0} := \frac{5}{α\sqrt{−11}} · Ω_{A}.

Computing this using MAGMA, we find that f(A, A[𝔑], ω_{A}) = −ᾱ^{−2} ×11. Since this last value is a 3-adic unit, Thm. 5.1.1 follows.

For Thm. 5.1.2, one needs to show that θ^{−3}f^{♭}(A, A[𝔑], ω_{A}) is not zero. Arguing as above, this value is congruent to δ^{1}f(A, A[𝔑], ω_{A}) modulo a prime above 3, so it suffices to check that δ^{1}f(A, A[𝔑], ω_{A}) is a 3-adic unit; indeed, we find computationally that δ^{1}f(A, A[𝔑], ω_{A}) = −ᾱ^{−3}×2^{2}×11. (One needs to justify that this value can be provably identified by giving an argument analogous to that for δ^{0}f(A, A[𝔑], ω_{A}) given above, that used the results of [DR]. For this justification, see the remarks following the end of the proof; this rests on Appendix B.) This shows that the rank of the group Gr^{3}(X_{2,2}) over K is at least 1, but now using the CM action of K on either of the elliptic curve factors of W_{2,2} (as in Sec. 4.3), we see that the ranks of both the ±1-eigenspaces for complex conjugation are at least one. ■

We now explain how one can provably compute δ^{1}f(A, A[𝔑], ω_{A}) or equivalently δ_{4}f(τ). First we use

the following idea due to Shimura (see [Shim], §1). Pick an element α = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL_{2}(Q) such that

α · τ = τ and α ≠ ±1. Let

h(z) := \frac{(f|_{4}α)(z)}{f(z)}.

(Here as usual F |_{ℓ}β(z) := F (β · z) · det(β)^{ℓ/2} · (cz + d)^{−ℓ}.) Then h(τ) = λ^{−4}, where λ := cτ + d. Applying δ_{4} to the relation

f|_{4}α = fh

yields

(δ_{4}f)|_{6}α = δ_{4}(f|_{4}α) = δ_{4}f · h + fδ_{0}h.

Evaluating at τ yields

δ_{4}f(τ) · λ^{−6} = δ_{4}f(τ) · λ^{−4} + f(τ)δ_{0}h(τ),

that is

λ^{−4}(λ^{−2} − 1) · δ_{4}f(τ) = f(τ) · δ_{0}h(τ).

So to compute δ_{4}f(τ), given that we’ve already computed f(τ), it suffices to bound the denominator of the algebraic part of δ_{0}h(τ).

In our case, we can take α to correspond (for example) to the image of \frac{1+\sqrt{−11}}{1−\sqrt{−11}} under the embedding K ↪ M_{2}(Q) associated with the point τ. Then

α = (\begin{matrix}−4/3 & −1/3 \\ 5/3 & −1/3 \end{matrix}) = \frac{1}{3}β

with

β := (\begin{matrix}−4 & −1 \\ 5 & −1 \end{matrix}) .

Note that γ := (\begin{matrix}−1 & 1 \\ −5 & 4 \end{matrix}) ∈ Γ_{0}(5) and

γ · β = (\begin{matrix}1 & −2 \\ 0 & 9 \end{matrix}) ,

so that

(f|_{4}α)(z) = (f|_{4}β)(z) = (f|_{4}γβ)(z) = 9^{−2} · f (\frac{z − 2}{9}) ,

and

h(z) = 9^{−2} · f (\frac{z − 2}{9}) · f(z)^{−1}.

Letting g(z) := 9^{2} · h(9z + 2), we get

g(z) = f(z) · f(9z + 2)^{−1} = f(z)/f(9z).

Now, δ_{0}h(τ) = 9^{−3} · δ_{0}g(τ′) , with τ′ := (τ − 2)/9. But g is a meromorphic modular function on Γ_{0}(9 · 5), hence δ_{0}g is a meromorphic modular form of weight 2 on Γ_{0}(9 · 5) and is given by the formula

δ_{0}g(z) = \frac{f(9z)f′(z) − 9f′(9z)f(z)}{f(9z)^{2}}.

Thus

F (z) := f(9z)^{2} · δ_{0}g(z) = f(9z)f′(z) − 9f′(9z)f(z)

is a holomorphic cusp form of weight 10 on Γ_{0}(3^{2} · 5). The algebraic part of F (τ′) is integral outside of the primes 3 and 5 and Theorem B.1.3.1 in Appendix B gives an effective lower bound on its denominator at 5. Tracing backwards through the computations above, we find that there is an effectively computable algebraic integer c_{1} ∈ 𝒪_{K} such that c_{1}δ^{1}f(A, A[𝔑], ω_{A}) is integral outside of 3. The argument above can now be repeated with a different choice of α, say corresponding to the image of \frac{5+3\sqrt{−11}}{5−3\sqrt{−11}}. (Note that the element (5 + 3\sqrt{−11})/2 has norm 31 which is the smallest split prime in K after 3 and 5.) In exactly the same way, we find that there is an effectively computable algebraic integer c_{2} ∈ 𝒪_{K} such that c_{2}δ^{1}f(A, A[𝔑], ω_{A}) is integral outside of 31. Then c_{1}c_{2}δ^{1}f(A, A[𝔑], ω_{A}) is integral, and hence can be computed effectively, as desired.

Remark 5.1.4. The reader will find in Appendix A a table of values δ^{k}f(A, A[𝔑], ω_{A}) for k in the range 0 through 21, computed to many digits of accuracy. From this table, we see that instead of p = 3 and k = 0, 1, we could argue instead using the split prime p = 23 and the values corresponding to k = 20 and 19, provided we could show that these computed values are indeed correct. However, this seems to be a nontrivial issue since our method of proof above for k = 0, 1, namely translating the problem to computing the value of a holomorphic form with integral q-expansion and applying Theorem B.1.3.1 in Appendix B, does not seem to extend easily to larger values of k.

5.2. The Fermat quartic revisited. We end by using the methods of this paper to give another proof of Bloch’s theorem (see §1) that rank Gr^{2}(X) ≥ 1 for X = the Jacobian of the Fermat quartic.

Recall that X is isogenous over Q to the product A × A × A′, where A and A′ are the elliptic curves given by the equations

A : y^{2} = x^{3} − x, A′ : y^{2} = x^{3} + x.

The curves A and A′ have conductors 32 and 64 respectively, and both have CM by the ring of integers of K = Q(i). We will need to identify explicitly the associated Grössencharacters ψ and ψ′ of K. Let 𝔮 = (1 + i) be the unique prime of 𝒪_{K} above 2, so that 𝔮^{2} = (2). Since the conductor N_{A} = N_{K/Q}(𝔠_{ψ})D and likewise for A′, the conductors of ψ and ψ′ must be 𝔮^{3} and 𝔮^{4} respectively. Further, since ψ and ψ′ are associated with elliptic curves, their restrictions to Q must equal ε_{K}. Since K has class number one, we have

A_{K}^{×} = K^{×} · (∏_{v} 𝒪_{v}^{×} × K_{∞}^{×})

and to give a Hecke character η = ∏_{v} η_{v} of conductor dividing 𝔮^{n} of infinity type (1, 0) is equivalent to giving a finite order character η_{𝔮} of U_{𝔮,n} := 𝒪_{𝔮}^{×}/1 + 𝔮^{n}𝒪_{𝔮} = (1 + 𝔮𝒪_{q})/(1 + 𝔮^{n}𝒪_{q}) such that for any global unit u ∈ 𝒪_{K}, we have

η_{𝔮}(u) = u.

Now U_{𝔮,3} ≃ Z/4Z is cyclic, generated by i, hence there is a unique character ψ of type (1, 0) and conductor 𝔮^{3}. Globally, this character is determined by

ψ((α)) = α, if α ∈ 𝒪_{K}, α ≡ 1 mod 𝔮^{3}.

Likewise U_{𝔮,4} ≃ Z/4Z×Z/2Z, where the first factor is generated by a := 2+i and the second by b := 1−2i. It is easy to see then that the only characters of conductor dividing 𝔮^{4} of infinity type (1, 0) are ψ and ψ′, the latter given by

ψ′_{𝔮}(a) = i, ψ′_{𝔮}(b) = 1.

In contrast, ψ satisfies

ψ_{𝔮}(a) = −i, ψ_{𝔮}(b) = −1.

Thus ψ′ = ψμ, where μ is the finite order (quadratic) character of K of conductor 𝔮^{4}, characterized by

μ_{𝔮}(a) = −1, μ_{𝔮}(b) = −1,

so that μ is in fact the quadratic character associated with the extension Q(ζ_{8})/K.

By the Künneth formula, the motive H^{3}(X) splits as the sum M_{θ_{χ}} ⊕ M_{θ_{ψ′}} (−1)^{3}, where χ is the Hecke character

χ := ψ^{2}ψ′ = ψ^{3}μ.

Let f be the modular form

f := θ_{ψ^{2}} = ∑_{𝔞} ψ^{2}(𝔞)e^{2πiN𝔞z}.

Since ψ^{2} has conductor 𝔮^{2}, the form f lies in S_{3}(Γ_{0}(16), ε_{K}). Let W be the elliptic modular surface over the modular curve X_{Γ} associated with the subgroup Γ := Γ_{0}(16) ∩ Γ_{1}(4) ⊂ SL_{2}(Z). By Künneth, H^{3}(W × E′) contains as a subfactor the motive

M_{f} ⊗ M_{θ_{ψ′}} = M_{θ_{ψ^{2}}} ⊗ M_{θ_{ψ′}} = M_{θ_{χ}} ⊕ M_{θ_{ψ′}} (−1).

Note that the group Γ is conjugate in SL_{2}(Z) to the subgroup Γ(4), so that W is isogenous to the (universal) elliptic modular surface W′ over X(4). (This isogeny will be described in moduli-theoretic terms below.) We will show rank Gr^{2}(X) ≥ 1 in two steps:

(i) Construct a generalized Heegner-type cycle Δ̃ on W′ × A′′, where A′′ is a curve Q-isogenous to A′ and show that Δ̃ is nontorsion in Gr^{2}(W′ × A′′).

(ii) Construct a correspondence Π from X to the product W′ × A′′, and verify that Δ̃ transfers via

Π to a cycle on X that is nontorsion in Gr^{2}(X). We start with (i). The curve X(4) has a canonical model over K, given as the solution to a fine moduli

problem, namely classifying triples (E, P, Q) where E is an elliptic curve, P and Q are points of order exactly 4, such that the Weil pairing ⟨P, Q⟩ = i. In fact X(4) is isomorphic to P^{1}, and an explicit equation for W′ is the classical Jacobi quartic

(5.2.1)  Y^{2} = (X^{2} − σ^{2})(X^{2} − σ^{−2}),

or in Weierstrass form:

(5.2.2)  Y^{2} = X(X − 1)(X − λ),

where

λ = \frac{1}{4}(σ + σ^{−1})^{2},

with σ being the parameter on X(4) = P^{1}. Note that X(4) and W′ are in fact defined over Q. A basis for the 4-torsion (in the Weierstrass form) is given by (P_{σ}, Q_{σ}), where

P_{σ} = (\frac{1}{2}(σ^{2} + 1)/σ^{2}, −\frac{1}{4}(σ^{4} − 1)/σ^{3});
Q_{σ} = (−\frac{1}{2}(σ^{2} + 1)/σ, \frac{1}{4}i(σ^{4} + 2σ^{3} + 2σ^{2} + 2σ + 1)/σ^{2}).

These satisfy ⟨P_{σ}, Q_{σ}⟩ = i, with ⟨·, ·⟩ being the Weil pairing.

Fix ζ a primitive 8th root of unity such that ζ^{2} = i. The fiber of W′ over σ = ζ is an elliptic curve isomorphic to the curve

A′′ : y^{2} = x^{3} − 4x

which is 2-isogenous to A′. On the variety W′ × A′′, let Δ ∈ CH^{2}(W′ × A′′)_{0} be the cycle obtained by applying the projector ϵ_{W′}ϵ_{A′′} to the diagonal in A′′ × A′′ ⊂ W′ × A′′. This cycle is defined over Q(ζ). Define

Δ̃ := Tr_{Q(ζ)/K}(Δ).

Then Δ̃ is defined over K and we will show below that it is nontorsion in Gr^{2}(W′ × A′′).

To verify the last claim, we use that there is an isogeny W′ → W over X(4) ≃ X_{Γ}. Since

( \begin{matrix}4 & 0 \\ 0 & 1 \end{matrix} )^{−1} Γ(4) ( \begin{matrix}4 & 0 \\ 0 & 1 \end{matrix} ) = Γ,

the natural isomorphism X(4) = Γ(4)\ℌ ≃ Γ\ℌ = X_{Γ} is given by z ↦ z/4. In terms of the moduli interpretations, this isomorphism sends a triple (E, P, Q) to the triple (E′, φ(P), φ(C)), where

φ : E → E′ := E/⟨Q⟩

is the natural isogeny, and C is the subgroup

C := {R ∈ E : 4R ∈ ⟨P ⟩}.

Thus C is a subgroup of E of order 64 but φ(C) is cyclic of order 16. The isogenies φ above patch together to give the desired isogeny from W′ to W over X(4) ≃ X_{Γ}. Further, under this last isomorphism, the modular form f on Γ is identified with the modular form η(z)^{6} on Γ(4), i.e., one has the identity:

f(z/4) = η(z)^{6},  or  f(z) = η(4z)^{6}.

We can now show:

Proposition 5.2.1. The cycle Δ̃ is nontorsion in Gr^{2}(W′ × A′′).

Proof. It suffices to show that the pushforward φ_{∗}(Δ̃) of Δ̃ under (φ, 1) : W′ ×A′′ → W ×A′′ is nontorsion in Gr^{2}. Let x_{ζ} denote the point of X(4) corresponding to σ = ζ and let y_{ζ} be its image in X_{Γ}. The point y_{ζ} then corresponds to a triple of the form (A′′, P, C) with P a point of order 4 on A′′ and C a cyclic subgroup of order 16 on A′′ containing P, both P and C being defined over Q(ζ). Let us fix a Néron differential ω on A′′ defined over Q. Using directly Lemma 3.22 and 3.23 and Prop. 3.24 of [BDP1], it suffices to show that

θ^{−2}f^{♭}(A′′, ω, P, C) + θ^{−2}f^{♭}(A′′, ω, P^{τ} , C^{τ}) ≠ 0,

where τ is the nontrivial Galois automorphism of Q(ζ)/K. Again, as in the previous section, it suffices to find a prime p split in K, say p = 𝔭𝔭̄, and a positive integer k ≡ −2 mod p − 1, such that

(5.2.3)  δ^{k}f(A′′, ω, P, C) + δ^{k}f(A′′, ω, P^{τ} , C^{τ}) ≢ 0 mod 𝔭.

Now, a simple computation shows that in the complex analytic uniformization, the point x_{ζ} corresponds to the class [i] ∈ Γ(4)\ℌ while its conjugate by τ corresponds to the point [γi] ∈ Γ(4)\ℌ, where

γ := ( \begin{matrix}−3 & −2 \\ 2 & 1 \end{matrix} ) .

Let Ω be the real period of A′′. The period lattice of A′′ is Ω · (Z + Zi) and γ · i = (i − 8)/5, so

(5.2.4)  δ^{k}f(A′′, ω, P, C) = (\frac{2πi}{Ω})^{3+2k} · δ^{k}f(i/4)

and

(5.2.5)  δ^{k}f(A′′, ω, P^{τ} , C^{τ}) = (\frac{2πi}{(2i + 1) · Ω})^{3+2k} · δ^{k}f((i − 8)/20)

These values can be computed, and we find that in fact the values of (5.2.4) and (5.2.5) are equal for all k and zero for odd values of k; for even k, a table of these values can be found in Appendix A. Taking either k = 2, p = 5 or k = 8, p = 11 or k = 14, p = 17, we see that (5.2.3) holds. ■

As in the previous section, one needs to establish that the computed values of δ^{k}f used above are indeed correct. In this case, the method of the previous section seems difficult to apply (since we need k ≥ 2), but one can argue differently as follows. Let

Λ^{±}(k) := δ^{k}f(A′′, ω, P, C) ± δ^{k}f(A′′, ω, P^{τ} , C^{τ}).

Under the embedding of K into M_{2}(Q) corresponding to z := i/4, the order

\{( \begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix} ) ∈ M_{2}(Z) : c′ ≡ 0 mod 16\}

pulls back to the order 𝒪_{c} in K of conductor c := 4. Since K(ζ) is the ring class field of K of conductor 4, by a formula of Waldspurger [Wa] one has (for k ≥ 0)

Λ^{+}(k)^{2} = c_{1}(k) · L^{alg}(f, ψ′ψ^{2k+2}, k + 3) = c_{1}(k) · L^{alg}(ψ′ψ^{2k+4}, k + 2) · L^{alg}(ψ′ψ^{2k}, k + 1)
= c_{1}(k) · L^{alg}(ψ^{2k+5}μ, k + 2) · L^{alg}(ψ^{2k+1}μ, k + 1),

and

Λ^{−}(k)^{2} = c_{2}(k) · L^{alg}(f, ψ′ψ^{2k+2}μ, k + 3) = c_{2}(k) · L^{alg}(ψ′ψ^{2k+4}μ, k + 2) · L^{alg}(ψ′ψ^{2k}μ, k + 1)
= c_{2}(k) · L^{alg}(ψ^{2k+5}, k + 2) · L^{alg}(ψ^{2k+1}, k + 1),

where c_{1}(k) and c_{2}(k) are elements in K that are possibly zero. (There is an exact version of Waldspurger’s formula worked out in [BDP1], §4 which does not quite cover this case since we assumed in that article that the imaginary quadratic field has odd discriminant and the conductor of the embedding K⟶M_{2}(Q) is also odd. However, for the current application, an exact formula is not needed.) The signs of the L-functions L(ψ^{2k+1}, s) and L(ψ^{2k+1}μ, s) are easy to compute; one finds that

sign L(ψ^{2k+1}, s) = \{ \begin{matrix}+1, if 2k + 1 ≡ 1, 3 mod 8; \\ −1, if 2k + 1 ≡ 5, 7 mod 8, \end{matrix}

and

sign L(ψ^{2k+1}μ, s) = \{ \begin{matrix}+1, if 2k + 1 ≡ 1, 5 mod 8; \\ −1, if 2k + 1 ≡ 3, 7 mod 8. \end{matrix}

It follows that simply for sign reasons, Λ^{−}(k) is always zero while Λ^{+}(k) is zero for odd values of k. For even k, at least in the range of computation of the table in Appendix A, one finds computationally that Λ^{+}(k) ≠ 0, hence also c_{1}(k) ≠ 0. Now c_{1}(k) is a product of local factors c_{1}(k) = ∏_{v} c_{1,v}(k), over all the places v of Q. The factor c_{1,v}(k) for v ≠ 2 are computed explicitly in [BDP1] §4, which as mentioned above does not cover the computation of c_{1,2}(k). However, c_{1,2}(k) is a local 2-adic integral and so one can effectively bound it from below in terms of the ramification data at 2, since one knows that is not zero. Then, as in §4.3, one can use [GS] Prop. 10.11 (a) to bound the denominator of L^{alg}(ψ^{2k+1}μ, k + 1) and hence the denominator of δ^{k}f(A′′, ω, P, C).

Remark 5.2.2. The fact that Λ^{−}(k) = 0 for all k implies that the image of Δ − Δ^{τ} under the p-adic Abel-Jacobi map is 0, at least for all primes p split in K. This is consistent with the BB-conjecture since Δ − Δ^{τ} lies in the −1-eigenspace of CH^{2}(W′ × A′′)_{0} for the action of Gal(Q(ζ)/K). The dimension of this eigenspace should be controlled by the order of vanishing at the center of the twist by μ of the motive M_{f} ⊗ M_{θ_{ψ′}}, namely the L-function

L(χμ, s)L(ψ′μ, s − 1) = L(ψ^{3}, s) · L(ψ, s).

This last L-function can be checked to be nonvanishing at the center.

Remark 5.2.3. As in the previous sections, the CM action of K on A′′ can be used to define cycles Δ̃^{±} in the respective ±1-eigenspaces for the action of complex conjugation, that are both nontorsion in Gr^{2}(W′ × A′′).

We now show (ii), by constructing an explicit correspondence Π_{1} : A × A −−→ W′ defined over Q. (This construction is likely classical, but we include it for want of a reference; see also [Schap], Remark 2.4.1.) First map A × A to A_{0} × A_{0}, where A_{0} is the Q-isogenous curve

A_{0} : y^{2} = x^{4} − 1.

Taking two copies of A_{0}, say y_{1}^{2} = x_{1}^{4} − 1, y_{2}^{2} = x_{2}^{4} − 1, define a map

A_{0} × A_{0} → W′,

the latter given by (5.2.1), by the formulas

(x_{1}, y_{1}, x_{2}, y_{2}) ↦ (x, y, σ)

where x = x_{1}x_{2}, y = y_{1}y_{2}, σ = x_{1}/x_{2}. It is straightforward to check that the unique (up to scaling) nonzero holomorphic 2-form ω_{f} on W′ pulls back under this map to a nonzero multiple of p_{1}^{∗}(ω_{A_{0}}) ∧ p_{2}^{∗}(ω_{A_{0}}), where ω_{A_{0}} is a nonzero holomorphic one form on A_{0}. The correspondence Π_{1} induces a correspondence Π : X −−→ A × A × A′ −−→ W′ × A′′, such that ω_{f} ∧ ω_{A′} pulls back to a nonzero multiple of the unique holomorphic 3-form on X, from which it follows immediately that the cycles Π^{∗}(Δ̃^{±}) are nontorsion in Gr^{2}(X).

Appendix A. Tables

Let L(k) := L^{alg}(θ_{ψ^{2k+1}}). These tables list the L-values L(k)^{1/2}, for a range of values of k.

\begin{matrix}k & 0 & 2 & 4 & 6 & 8 \\ L(k)^{1/2} & 1 & 2 & 2 & 6 & 14 \end{matrix}

Table 1: L(k)^{1/2} for D = 7, 0 ≤ k ≤ 8, k even.

k\D  11  19  43  67
1  1  1  1  1
3  2  2 · 3  2^{2} · 3^{2}  2 · 3 · 19
5  2^{3}  2^{4}  2^{3} · 5 · 11  2^{3} · 5 · 173
7  2 · 7  2 · 3 · 31  2 · 3^{2} · 7 · 151  2 · 3 · 7 · 61^{2}
9  2^{4} · 19  2^{4} · 3^{2} · 29  2^{5} · 3^{2} · 29  2^{4} · 3^{3} · 17^{2} · 29^{2}
11  2^{5} · 11  2^{6} · 3^{2} · 59  2^{5} · 3^{6} · 5 · 17^{2}  2^{5} · 3^{2} · 5 · 11 · 251969
13 2^{3} · 11 · 13 · 67  2^{3} · 3 · 13 · 1747 2^{3} · 3 · 19 · 653 · 6967  2^{3} · 3 · 13 · 150044563
15 2^{4} · 7 · 11 · 223 2^{4} · 3^{2} · 23 · 83 · 149  2^{5} · 3^{3} · 5 · 7 · 11  2^{4} · 3^{2} · 5 · 7 · 19
191 · 751  ·79 · 181 · 66359
17  2^{8} · 11 · 17^{2}  2^{9} · 3 · 203911 2^{8} · 3 · 5 · 1877273779 2^{8} · 3 · 5 · 4639448759053
19  2^{6} · 11 · 19  2^{6} · 3^{3} · 19  2^{6} · 3^{4} · 19 · 29  2^{6} · 3^{3}
·47 · 109  ·29 · 179 · 223  ·37 · 7247281  ·690894919089413
21  2^{8} · 7 · 11  2^{8} · 3^{3} · 19  2^{9} · 3^{3} · 5 · 7 · 127  2^{8} · 3^{3} · 5 · 7 · 19 · 109
·317 · 347  ·2318077  ·257 · 10151899  793960993361
Table 2: L(k)^{1/2} for D = 11, 19, 43, 67, 1 ≤ k ≤ 21, k odd.

\begin{matrix}k\D & 163 \\ 1 & 1 \\ 3 & 2^{2} · 3 · 181 \\ 5 & 2^{3} · 5^{2} · 17569 \\ 7 & 2 · 3 · 7 · 523 · 152389 \\ 9 & 2^{5} · 3^{2} · 829366883 \\ 11 & 2^{5} · 3^{2} · 5^{2} · 11 · 1213 · 1579 · 25981 \\ 13 & 2^{3} · 3 · 13 · 17 · 53 · 6775046888887 \\ 15 & 2^{5} · 3^{2} · 5 · 7 · 61 · 28499 · 472933737329 \\ 17 & 2^{8} · 3 · 5^{4} · 17 · 1012119331 · 2184243869 \\ 19 & 2^{6} · 3^{3} · 19 · 2347 · 122795513 · 235220529149 \\ 21 & 2^{9} · 3^{3} · 5 · 7 · 43 · 701 · 685301 · 10549227661986929 \\ 23 & 2^{9} · 3^{3} · 5^{2} · 7 · 11 · 23 · 584359 · 17638423193 · 46435917679 \\ 25 & 2^{7} · 3^{2} · 5^{2} · 373 · 17747 · 11959219 · 61407268956238912073 \\ 27 & 2^{9} · 3^{5} · 5^{2} · 13 · 6551 · 16995477783287 · 510845704143990001 \\ 29 & 2^{10} · 3^{4} · 5^{3} · 7 · 29 · 1907 · 602217754893611 · 4889245158810100367 \\ 31 & 2^{8} · 3^{5} · 5 · 7 · 17 · 31 · 491 · 4345577 · 11642395177462816343370659054897 \\ 33 & 2^{14} · 3^{5} · 5 · 11 · 14895653 · 7649341745993183353696263632853979211 \\ 35 & 2^{14} · 3^{5} · 5^{3} · 7 · 11 · 17 · 233 · 281 · 1454839941553 · 9691613471414784647219053009 \\ 37 & 2^{12} · 3^{4} · 5 · 7 · 37 · 67 · 199 · 51239 · 13274045931043226796915472627219738949574163 \\ 39 & 2^{14} · 3^{5} · 5 · 7 · 13 · 19 · 43 · 53 · 79 · 1344250377771491844069374345645234060030577543203 \\ 41 & 2^{16} · 3^{4} · 5^{3} · 13 · 2027 · 2197653300070968660189451708159964098498078082797912571 \end{matrix}

Table 3: L(k)^{1/2} for D = 163, 1 ≤ k ≤ 41, k odd.

\begin{matrix}k & ᾱ^{k+2} · δ^{k}f(A, A[𝔑], ω_{A}) \\ 0 & −11 \\ 1 & −2^{2} · 11 \\ 2 & −2^{2} · 5 · 11 \\ 3 & +2 · 11 · 61 \\ 4 & +2^{3} · 11 · 137 \\ 5 & +2^{5} · 7^{2} · 11 · 13 \\ 6 & −2^{4} · 5 · 11 · 17 · 137 \\ 7 & +2^{5} · 7 · 11 · 5407 \\ 8 & −2^{6} · 11^{2} · 31 · 7639 \\ 9 & +2^{6} · 11^{2} · 295459 \\ 10 & −2^{7} · 5 · 11^{2} · 1592281 \\ 11 & +2^{10} · 7 · 11^{2} · 149 · 199 · 929 \\ 12 & −2^{6} · 11^{2} · 61126101181 \\ 13 & −2^{10} · 7 · 11^{2} · 13 · 41 · 151 · 52609 \\ 14 & −2^{10} · 5^{2} · 7 · 11^{2} · 10163 · 1317713 \\ 15 & +2^{9} · 7 · 11^{2} · 23 · 17293 · 7958443 \\ 16 & −2^{13} · 11^{2} · 365267262447743 \\ 17 & +2^{13} · 11^{2} · 17 · 547 · 3491 · 374761997 \\ 18 & +2^{13} · 5 · 7 · 11^{2} · 173 · 1213 · 103339254493 \\ 19 & −2^{15} · 7 · 11^{3} · 13 · 19 · 289837 · 788970227 \\ 20 & −2^{13} · 11^{3} · 141023 · 1727597 · 338696707 \\ 21 & +2^{14} · 7^{2} · 11^{3} · 67 · 16427 · 36677 · 266115323 \end{matrix}

Table 4: ᾱ^{k+2} · δ^{k}f(A, A[𝔑], ω_{A}) for 0 ≤ k ≤ 21 f ∈ S_{4}(Γ_{0}(5)); notations as in Sec. 5.1.

\begin{matrix}k & 2^{−(3+3k)} · i^{−1} · δ^{k}f(A′′, ω, P, C) \\ 0 & −1 \\ 2 & +1 \\ 4 & +3 \\ 6 & −3 · 17 \\ 8 & −3^{2} · 17 \\ 10 & −3^{3} · 37 \\ 12 & +3^{2} · 7 · 37 · 113 \\ 14 & +3^{2} · 7 · 43 · 97 · 113 \\ 16 & −3^{4} · 43 · 97 · 3203 \\ 18 & −3^{5} · 3203 · 16231 \\ 20 & −3^{5} · 7 · 11 · 53 · 89 · 16231 \end{matrix}

Table 5: 2^{−(3+3k)} · i^{−1} · δ^{k}f(A′′, ω, P, C) for 0 ≤ k ≤ 20, k even f ∈ S_{3}(Γ_{1}(16)), notations as in Sec. 5.2.

Appendix B. Integrality of modular forms via intersection theory by Brian Conrad

B.1. Introduction.

B.1.1. Integrality and q-expansions. Let k, N ≥ 1 be integers, and Γ = Γ_{0}(N). For a weight-k modular form f ∈ M_{k}(Γ, C) on Γ, let f_{∞} ∈ C[[q]] denote its q-expansion at ∞. A general argument unrelated to modular curves (see (B.1.2.1) and Proposition B.2.1.1) ensures that for any number field K ⊂ C, the 𝒪_{K}-module

M_{k}(Γ, 𝒪_{K}) := {f ∈ M_{k}(Γ, C) | f_{∞} ∈ 𝒪_{K}[[q]]}

is finitely generated and is an 𝒪_{K}-structure on M_{k}(Γ, C) (in the sense that C⊗_{𝒪_{K}} M_{k}(Γ, 𝒪_{K}) → M_{k}(Γ, C) is an isomorphism). This 𝒪_{K}-structure is useful because membership in M_{k}(Γ, 𝒪_{K}) can be verified in practice via Riemann–Roch arguments by computing enough q-expansion coefficients. (The p-integrality aspects for primes p|N rest on the geometry of the irreducible component through ∞ in the moduli stack X_{0}(N)_{F_{p}}; see Definition B.1.2.1.)

This 𝒪_{K}-structure suffers from a serious technical drawback, as follows. Let E be an elliptic curve over K with good reduction over 𝒪_{K} and Néron model ℰ (the cases of most interest being CM elliptic curves), and let G ⊂ E be a cyclic K-subgroup of order N. Choose a nonzero ω ∈ Ω^{1}(E). For a modular form f ∈ M_{k}(Γ, 𝒪_{K}) and the point x ∈ X_{0}(N)(K) arising from (E, G), the evaluation f(x) lies in Ω^{1}(E)^{⊗k} but may fail to lie in the 𝒪_{K}-line Ω^{1}(ℰ)^{⊗k} (even though f has q-expansion coefficients in 𝒪_{K}); we only know that f(x) ∈ Ω^{1}(ℰ)^{⊗k}[1/N]. If we want to rigorously justify an evaluation of f(x)/ω^{⊗k} ∈ K on a computer then we need a theoretical multiplicative upper bound on the the “denominator” of f(x) with respect to the 𝒪_{K}-line Ω^{1}(ℰ)^{⊗k}. This is a question concerning powers of the primes p|N.

There is a finite-index 𝒪_{K}-submodule M_{k,𝒪_{K}} ⊆ M_{k}(Γ, 𝒪_{K}) with the property that f(x) ∈ Ω^{1}(ℰ)^{⊗k} for any f ∈ M_{k,𝒪_{K}} . This 𝒪_{K}-structure on M_{k}(Γ, C) is defined in terms of a regular proper moduli stack over Z, as we shall review in §B.1.2. There are compatible equalities

𝒪_{K} ⊗_{Z} M_{k,Z} = M_{k,𝒪_{K}} , 𝒪_{K} ⊗_{Z} M_{k}(Γ, Z) = M_{k}(Γ, 𝒪_{K}),

so the exponent of the finite abelian group M_{k}(Γ, Z)/M_{k,Z} multiplies M_{k}(Γ, 𝒪_{K}) into M_{k,𝒪_{K}} and hence bounds all failure of integrality when evaluating elements of M_{k}(Γ, 𝒪_{K}) at CM points over K. This motivates our aim in this appendix: to compute an explicit nonzero multiple of the exponent of M_{k}(Γ, Z)/M_{k,Z}.

Remark B.1.1.1. The exponent of M_{k}(Γ, Z)/M_{k,Z} is divisible by all primes p|N when k is sufficiently large and even; see Corollary B.3.1.5. More specifically, for even k ≫_{N} 1 there exists f ∈ M_{k}(Γ, Z) such that for each p|N there is a cusp at which the q-expansion of f is not p-integral.

The case ord_{p}(N) = 1 with k = 2 is thoroughly analyzed in [E, §2] via the geometry of Néron models of Jacobians, building on ideas introduced in the case N = p for any k in [DR, VII, 3.19–3.20]. The technique used by Deligne and Rapoport rests on intersection theory on the regular proper Deligne–Mumford stack X_{0}(p) over Z_{(p)}, yielding that p^{⌈kp/(p−1)⌉} is a multiple of the exponent in such cases. The arguments in [DR, VII, 3.19] use that X_{0}(p)_{F_{p}} is reduced with exactly two irreducible components. For general levels, we have to overcome the failure of reducedness and especially failure of the Deligne–Mumford condition for X_{0}(N)_{F_{p}} when ord_{p}(N) > 1.

Although the intersection theory will take place on a regular proper Artin stack over Z that is not Deligne–Mumford when N is not squarefree, this stack has finite automorphism schemes at its geometric points. We are not aware of any literature reference for intersection theory on Artin stacks beyond the Deligne–Mumford case, but the stacks that arise for us have additional special properties that will enable us to develop the required intersection theory by bootstrapping from the scheme case.

B.1.2. Moduli stacks. To formulate our results in a precise form, we need to define the 𝒪_{K}-structure M_{k,𝒪_{K}} mentioned above. This rests on the following:

Definition B.1.2.1. Let X = X_{0}(N) be the moduli stack over Spec(Z) that classifies generalized elliptic curves E → S (over arbitrary base schemes S) equipped with a Γ_{0}(N)-structure: a relatively ample finite locally free closed subgroup scheme G ⊂ E^{sm} that is “cyclic” in the sense that it admits a Drinfeld Z/NZ-generator fppf-locally on S.

See [Co, §2.3] (especially Definition 2.3.4 there) for the notion of cyclicity in Definition B.1.2.1. Since we are interested in integrality properties of modular forms, we must work over Z and not just over Z[1/N]. By [Co, Thm. 1.2.1], X is a regular proper flat Artin stack over Z with geometrically connected fibers of pure dimension 1 and it is smooth over Z[1/N].

It is essential that we consider the moduli stack X rather than its associated coarse space, since modular forms for Γ = Γ_{0}(N) correspond to sections of a line bundle on X that generally does not descend to the coarse space. The stack X is never a scheme, but it is Deligne–Mumford precisely on the open substack complementary to specific cusps that (always) exist in characteristics p for which p^{2}|N [Co, Thm. 3.1.7].

Let π : E → X be the universal generalized elliptic curve over the moduli stack, and let ω_{E/X} be its relative dualizing sheaf, so

ω := π_{∗}(ω_{E/X})

is a line bundle on X [DR, II, 1.6]. For k ≥ 1, the torsion-free Z-module of global sections M_{k,Z} := H^{0}(X, ω^{⊗k}) is finitely generated (by pullback to a Z-flat finite scheme covering defined via auxiliary level structure, or by coherence of higher direct images for proper Artin stacks [Ol]). This Z-module is a Zstructure on the C-vector space M_{k}(Γ, C) of weight-k classical modular forms for Γ since H^{0}(X_{C}, ω^{⊗k}) = M_{k}(Γ, C) (via GAGA applied to a higher-level moduli problem that is “rigid” and in fact represented by a scheme).

Since 𝒪_{K} is Z-flat, clearly

M_{k,𝒪_{K}} := H^{0}(X_{𝒪_{K}} , ω^{⊗k}) = 𝒪_{K} ⊗_{Z} M_{k,Z}.

We want to relate M_{k,𝒪_{K}} to the 𝒪_{K}-submodule M_{k}(Γ, 𝒪_{K}) ⊆ M_{k}(Γ, C) defined in §B.1.1 via the analytic theory of q-expansions. For this purpose, we now recall how q-expansions are defined algebraically in terms of the “completion” along a suitable Z-point of the stack X.

Let X^{∞} ⊂ X be the closed cuspidal substack that is the “locus of non-smoothness” of the universal generalized elliptic curve E → X [Co, Def. 2.1.8, Thm. 2.1.12]. This is a relative effective Cartier divisor in X [Co, Thm. 4.1.1]. Define ∞ : Spec(Z) → X to be the morphism arising from the standard 1-gon equipped with the canonical inclusion of μ_{N} into its smooth locus G_{m}; this morphism is a degree-2 finite étale cover of a connected component ∞_{X} of the cuspidal substack of X, and ∞_{X} lies inside the open Deligne–Mumford locus in X (by [Co, Thm. 3.1.7]).

For the morphism h : Spec(Z[[q]]) → X corresponding to the Tate curve equipped with the standard μ_{N} subgroup inside its smooth locus, the induced morphisms Spec(Z[q]/(q^{n+1})) → X are compatibly identified with étale double covers of the infinitesimal neighborhoods of ∞_{X} in X. Pullback along h defines q-expansions in A ⊗_{Z} Z[[q]] ⊆ A[[q]] on M_{A} := H^{0}(X_{A}, ω^{⊗k}) for any ring A, and for A = C this coincides with the analytic theory of q-expansions.

To understand the 𝒪_{K}-module M_{k}(Γ, 𝒪_{K}) in more geometric terms, let U ⊂ X be the open substack obtained by removing the fibral irreducible components of X → Spec Z disjoint from ∞_{X}, so U[1/N] = X[1/N] and U is a Z-smooth stack with geometrically irreducible fibers. The stack U is Deligne–Mumford [Co, Thm. 3.1.7], so by using h and the completion of an étale scheme neighborhood of ∞_{X} we see that the proof of the algebraic q-expansion principle in the “scheme” case applies to the Artin stack U. Thus, the inclusion

(B.1.2.1)  H^{0}(U_{𝒪_{K}} , ω^{⊗k}) ⊂ M_{k}(Γ, 𝒪_{K})

is an equality. In particular, since 𝒪_{K} ⊗_{Z} H^{0}(U, ω^{⊗k}) → H^{0}(U_{𝒪_{K}} , ω^{⊗k}) is an isomorphism (by Z-flatness of 𝒪_{K}), we see that the natural map 𝒪_{K} ⊗_{Z} M_{k}(Γ, Z) → M_{k}(Γ, 𝒪_{K}) is an isomorphism. Thus, the inclusion of finitely generated 𝒪_{K}-modules

H^{0}(X_{𝒪_{K}} , ω^{⊗k}) ⊂ M_{k}(Γ, 𝒪_{K})

has finite index (as both sides are 𝒪_{K}-structures on H^{0}(X_{K}, ω^{⊗k})) and it is obtained by scalar extension along Z → 𝒪_{K} of the analogous inclusion over Z. We conclude that the exponent of M_{k}(Γ, Z)/M_{k,Z} multiplies M_{k}(Γ, 𝒪_{K}) into M_{k,𝒪_{K}} .

B.1.3. Results. For the purpose of understanding the failure of integrality when evaluating elements of M_{k}(Γ, 𝒪_{K}) at CM-points, it suffices to solve:

Problem: Find an explicit multiple m(k, N) of the exponent of M_{k}(Γ_{0}(N), Z)/H^{0}(X_{0}(N), ω^{⊗k}).

It is natural to consider additional moduli problems such as Γ_{1}(N) and Γ(N). We will focus on the above Problem for Γ = Γ_{0}(N) and leave it to the interested reader to consider other cases. Since the moduli stack for Γ_{1}(Nqℓ) is a scheme for any two distinct primes q, ℓ ≥ 5 not dividing N, to merely find some theoretical multiplicative bound on the exponent for Γ_{0}(N) it suffices to consider moduli schemes with extra level structure. Although this bypasses the intervention of Artin stacks, it appears to yield much worse bounds on the exponent. Thus, we prefer to work throughout with the Z-stack X_{0}(N) (and its intersection-theoretic properties). The regularity of X_{0}(N) and the finiteness of the automorphism group schemes at its geometric points are crucial for the intersection theory arguments below. This regularity is generally lost upon passing to the coarse space.

Here is a special case of our main result, analogous to the result in [DR] for Γ_{0}(p):

Theorem B.1.3.1. For any integer k ≥ 1 and number field K, if m := ord_{p}(N) ∈ {1, 2, 3} then

p^{⌈kp^{m−1}(m+1/(p−1))⌉}M_{k}(Γ_{0}(N), 𝒪_{K}) ⊆ H^{0}(X_{0}(N)_{𝒪_{K}} , ω^{⊗k}).

In Theorem B.3.2.1 we address all m > 3 as well. For U_{p}-eigenforms of finite slope on Γ_{1}(N) (with m = ord_{p}(N) arbitrary), Bryden Cais used an argument with Igusa curves to establish a much better multiplier: p^{m(k−1)} works in such cases. However, it would be too restrictive for us to consider only U_{p}-eigenforms (e.g., F ∈ S_{10}(Γ_{0}(45)) in §5.1 does not appear to be an eigenform for any Hecke operator).

B.2. Intersection-theoretic bounds.

B.2.1. Elimination of q-expansions. The first step in the proof of Theorem B.1.3.1 and its generalizations is to reformulate the problem is purely geometric terms, without reference to modular curves or q-expansions. For the convenience of the reader, we first record a general geometric result that includes module-finiteness of M_{k}(Γ, 𝒪_{K}) = 𝒪_{K} ⊗_{Z} M_{k}(Γ, Z) as a special case. The localization M_{k}(Γ, Z)[1/N] = H^{0}(X_{Z[1/N]}, ω^{⊗k}) is Z[1/N]-finite, so to prove that M_{k}(Γ, Z) is Z-finite it suffices to check Z_{(p)}-finiteness of M_{k}(Γ, Z)_{(p)} for each of the finitely many primes p|N. This latter finiteness follows by applying the following to the open substack U ⊂ X_{0}(N) defined above (B.1.2.1) and to the invertible sheaf ω^{⊗k} on X_{0}(N)_{Z_{(p)}}:

Proposition B.2.1.1. Let (R, 𝔪) be a 1-dimensional noetherian local domain with fraction field F, and let X be a proper flat Artin stack over R with connected normal generic fiber. Let U ⊂ X be an open substack such that U_{F} = X_{F} and U → Spec(R) is surjective. For any vector bundle ℱ on X, ℱ (U) is finitely generated over R.

Proof. Since the inclusion U ↪ X becomes an equality after localization at F, general “spreading out” principles provide a nonzero a ∈ R such that U_{R[1/a]} = X_{R[1/a]}. Thus, ℱ (U)_{R[1/a]} = ℱ (U_{R[1/a]}) = ℱ (X_{R[1/a]}) is R[1/a]-finite, and R-flatness of ℱ implies that the localization maps ℱ (X) → ℱ (X)_{R[1/a]} and ℱ (U) → ℱ (U)_{R[1/a]} are injective. Hence, we have inclusions of R-modules

ℱ (X) ⊂ ℱ (U) ⊂ ℱ (X_{R[1/a]}),

and it suffices to find an n ≥ 0 such that ℱ (U) ⊂ a^{−n}ℱ (X) inside ℱ (X_{R[1/a]}). We may assume a ∉ R^{×}, so a ∈ 𝔪.

For m ≥ 0, define ℱ (X)_{m} to be the R-module of elements f ∈ ℱ (X) such that f|_{U} has vanishing image in Γ(U, ℱ /(a^{m})). We claim there exists n ≥ 0 so that ℱ (X)_{n} ⊆ aℱ (X). Granting this, let us see that ℱ (U) ⊂ a^{−n}ℱ (X). Pick f ∈ ℱ (U), so the equality ℱ (U)_{R[1/a]} = ℱ (X)_{R[1/a]} provides an exponent ν ≥ 0 such that a^{ν}f ∈ ℱ (X). If ν > n then a^{ν}f ∈ ℱ (X)_{n} ⊆ aℱ (X), so a^{ν−1}f ∈ ℱ (X). Continuing by descending induction, a^{n}f ∈ ℱ (X). This conclusion obviously also holds if ν ≤ n. Thus, f ∈ a^{−n}ℱ (X) in all cases, or in other words ℱ (U) ⊂ a^{−n}ℱ (X) as desired.

It remains to find an n so that ℱ (X)_{n} ⊆ aℱ (X). It suffices to prove the same after the faithfully flat extension R → R̂ to the completion of R (which may not be a domain if R is not normal, but in which a is not a zero-divisor), since (i) the analogous submodule ℱ (X)′_{n} over R̂ (i.e., the set of f ∈ ℱR̂(XR̂) such that f|_{UR̂} has vanishing image in Γ(UR̂, ℱR̂/(a^{m}))) is equal to R̂⊗_{R}ℱ (X)_{n}, and (ii) R̂⊗_{R}ℱ (X) = ℱ′(X′) for X′ := XR̂ and ℱ′ := ℱR̂.

Consider the descending intersection I = ⋂_{m≥1} ℱ (X)′_{m}. We shall prove that I = 0. Choose an element f ∈ I, so the restriction f|_{U′} over U′ = UR̂ vanishes along the a-adic completion of ℱ′ over a smooth scheme cover of U′. Thus, f|_{U′} vanishes over a Zariski-open neighborhood of the closed substack U′ ∩ {a = 0} that contains the special fiber of U′, so f|_{U} vanishes over an open substack V ⊂ U that contains the non-empty special fiber of U. It follows that V is non-empty, so V_{F} is a non-empty open locus inside the connected normal X_{F}. The locus of vanishing stalks for a global section of a vector bundle on a normal noetherian Artin stack is both open and closed, so by connectedness of X_{F} the global section f_{F} ∈ ℱ_{F} (X_{F}) vanishes and hence f = 0 (since X is R-flat). This proves that I = 0.

Chevalley proved that a decreasing sequence of submodules {M_{m}} of a finitely generated module M over a complete local noetherian ring B is cofinal for the max-adic topology on M if ⋂ M_{m} = 0. (This is proved for M = B in [Mat, Exer. 8.7], and the proof there applies to any finitely generated module.) Since R/aR is an artin local ring, there is an e ≥ 1 such that 𝔪^{e} ⊆ aR. By Chevalley’s result applied to M = ℱ (X)′ and M_{m} := ℱ (X)′_{m} over B = R̂, there is an n such that ℱ (X)′_{n} ⊆ 𝔪^{e}ℱ (X)′ ⊆ aℱ (X)′, so ℱ (X)_{n} ⊆ aℱ (X). ■

The inclusion M_{k,Z} := H^{0}(X, ω^{⊗k}) ⊆ H^{0}(U, ω^{⊗k}) = M_{k}(Γ, Z) between finite free Z-modules becomes an equality after inverting N. Thus, to construct an explicit multiplicative upper bound on the exponent of M_{k}(Γ, Z)/M_{k,Z}, for each prime p|N we seek an explicit exponent e_{p} ≥ 0 (depending on k, p, and N) so that p^{e_{p}}M_{k}(Γ, Z)_{(p)} ⊆ (M_{k,Z})_{(p)}. Just as Deligne and Rapoport solved this problem for Γ_{0}(p) by using intersection theory on the Deligne–Mumford stack X_{0}(p), we will use intersection theory on the Artin stack X_{0}(N). Thus, we now digress to develop a general formalism for intersection theory on a special class of regular proper flat Artin stacks over Dedekind domains (to be applied to X_{0}(N)).

B.2.2. Divisor and intersection theory formalism. Let X be a regular noetherian Artin stack that is proper and flat over a discrete valuation ring R with fraction field F, residue field k, and uniformizer π. Assume the following three properties: (i) X is Deligne–Mumford away from a nowhere-dense closed substack of the special fiber X_{0}, (ii) both fibers of X → Spec(R) have pure dimension 1, and (iii) there exists a regular finite flat scheme cover h : X′ → X such that X′ is also regular.

The existence of the regular finite flat scheme cover X′ may appear to be restrictive (when X is not a scheme), but it is satisfied by X_{0}(N); e.g., we can take X′ to be the moduli stack corresponding to Γ_{1}(Nqℓ) for distinct primes q, ℓ ≥ 5 not dividing N, with h the “forgetful” map that involves a contraction along the cusps. (This X′ is a regular scheme by [Co, Thm. 4.1.1(2), Thm. 4.2.1(2)], and the corresponding h is finite flat by [Co, Lemma 4.2.3(2),(4)]). Since h is a finite flat surjection, X′ has fibers of pure dimension 1 over Spec(R). Thus, the usual intersection theory of regular arithmetic surfaces applies to X′.

In this section we will use h-pullback to define an intersection theory formalism on X by pulling down the usual theory on X′ (and the resulting constructions will be independent of the choice of h). By applying this to X_{0}(N) we shall use the descriptions of local deformation rings in [KM, Ch. 13] to compute intersection numbers and multiplicities on X_{0}(N) without any interference from auxiliary level structures or isotropy groups.

Remark B.2.2.1. Assume X and X′ are connected. If X → Spec(R) is its own Stein factorization then it does not generally follow that X′ → Spec(R) is its own Stein factorization (though for X = X_{0}(N) and X′ = X_{1}(Nqℓ) each has structural map to Spec(Z) that is its own Stein factorization). Thus, it is important to note at the outset that the development of intersection theory on X′ as in [Ch, §2–§3] does not use Stein factorization, only the connectedness of X′. Since intersection theory on X′ relative to the base Spec(R) “makes sense” whenever the regular arithmetic surface X′ is merely connected, it is reasonable to use that theory to define an intersection theory on X.

In the scheme case, Stein factorization is essential for the proof that the intersection matrix among the irreducible components of the special fiber is negative semi-definite with 1-dimensional defect space [Ch, Lemma 7.1(b)]. Thus, we will need to be attentive to Stein factorization issues when we analyze the dimension of the defect space for the intersection matrix associated to the special fiber of the stack X.

Our task is essentially one of using descent theory to make definitions on X using the known formalism on X′. The main complications are related to connectedness: X′_{0} may have many more connected components than X_{0}, X_{0} can be disconnected when X is connected, and 𝒪(X′) may be larger than 𝒪(X) (so we cannot easily relate the Stein factorizations of X and X′ over Spec(R)). To handle issues related to Stein factorization later, we record:

Proposition B.2.2.2. Let X′ be a connected regular arithmetic surface over a henselian discrete valuation ring R. The intersection matrix associated to X′ → Spec(R) is negative semi-definite with a 1-dimensional defect space.

Proof. Let k be the residue field of R, and let X′ → Spec(R′) be the Stein factorization of X′. Here R′ = 𝒪(X′) has no nontrivial idempotents since X′ is connected, yet it is a finite product of local rings since it is module-finite over the henselian local R. Thus, R′ is local. The normality of X′ implies that R′ is normal [EGA, II, 8.8.6.1], so R′ is a discrete valuation ring. Letting π and π′ denote respective uniformizers of R and R′, the special fibers X′_{k} and X′_{k′} are identified with X′ mod π and X′ mod π′ respectively. Thus, X′_{k′} = X′_{k} mod \overline{π}′ where \overline{π}′ := π′ mod πR′, so X′_{k′} and X′_{k} have the same underlying reduced schemes.

It follows that the (reduced) irreducible components of X′_{k′} and X′_{k} are the same closed subschemes of X′, so any pair of distinct components has intersection scheme that is the same whether we view the base ring as R or R′. Hence, for such distinct components C_{1} and C_{2}, the intersection pairings (C_{1}, C_{2}) and (C_{1}, C_{2})′ relative to R and R′ are respectively

deg_{k}(C_{1} ∩ C_{2}) = [k′ : k] deg_{k′}(C_{1} ∩ C_{2}), deg_{k′}(C_{1} ∩ C_{2}).

The self-intersection numbers (C, C) and (C, C)′ for such components are determined by two data: (i) the respective pairings (C_{1}, C_{2}) and (C_{1}, C_{2})′ for C_{1} ≠ C_{2} and (ii) the multiplicity μ_{C} of the special fiber along each component C. But clearly ord_{C}(π) = e(R′|R)ord_{C}(π′) for every C, so the multiplicities all scale by the same factor e(R′|R) when passing between the base rings R and R′. Hence, if {C_{i}} is the common set of (reduced) irreducible components of the special fibers and μ_{i} := μ_{C_{i}} then

(C_{i}, C_{i}) = ∑_{j≠i} \frac{μ_{j}}{μ_{i}} (C_{j}, C_{i}) = ∑_{j≠i} \frac{e(R′|R)μ′_{j}}{e(R′|R)μ′_{i}} [k′ : k](C_{j}, C_{i})′ = [k′ : k](C_{i}, C_{i})′.

To summarize, the intersection matrix associated to the special fiber of X′ → Spec(R) is [k′ : k] times the analogous such matrix associated to X′ → Spec(R′). This latter matrix is negative semi-definite with a 1-dimensional defect space by [Ch, Lemma 7.1(b)], and scaling by [k′ : k] preserves these properties. ■

It is convenient at the start to define some notions for proper Artin stacks D of pure dimension 1 over an arbitrary field k (which can also be F = Frac(R) as above in subsequent applications), ignoring regularity and connectedness conditions. We assume that there is a finite flat scheme cover q : D′ → D. Note that for any z ∈ D(k), the k-morphism z : Spec(k) → D is finite because its q-pullback is q^{−1}(z) → D′ (with q^{−1}(z) a finite k-scheme). Thus, the automorphism group scheme Aut(z) at any k-point z of D is k-finite since it is the fiber product

Spec(k) ×_{z,D,z} Spec(k).

For each z ∈ D(k), we let # Aut(z) denote the order of the finite k-group scheme Aut(z).

Every connected component C′ of D′ is finite flat over a unique connected component C of D, and conversely every connected component C of D is the image of any connected component C′ of the open and closed subscheme q^{−1}(C) of D′. For such C and C′, let q_{C′} : C′ → C be the finite flat map induced by q; it has constant fibral degree.

Lemma B.2.2.3. Let ℒ be an invertible sheaf on D, and let Z ⊂ D be a nowhere-dense closed substack. For all connected components C′ of D′ lying over a given connected component C of D, the ratios

\frac{deg_{k}(q_{C′}^{∗}(ℒ |_{C}))}{deg(q_{C′})} , \frac{deg_{k}(q_{C′}^{−1}(Z ∩ C))}{deg(q_{C′})}

are independent of C′, and also are independent of the choice of D′.

We denote these ratios as deg_{k}(ℒ |_{C}) and deg_{k}(Z ∩ C) respectively.

Proof. The problem is intrinsic to the finite flat covering map q_{C′} (allowing variation in C′), so we may assume D and D′ are connected and just have to check that deg_{k}(q^{∗}(ℒ))/ deg(q) and deg_{k}(q^{−1}(Z))/ deg(q) are independent of the choice of q : D′ → D.

Let D′′ → D be another connected finite flat scheme cover, so D′ ×_{D} D′′ is finite flat over D′ and D′′ (in particular, this fiber product is a scheme). Any connected component of D′ ×_{D} D′′ is a finite flat cover of D′ and D′′ (as D′ and D′′ are connected), so we may rename such a connected component as D′′ to reduce to the case that D′′ → D is the composition of q with a finite flat scheme surjection q′ : D′′ → D′. Connectedness of D and D′ ensures that the degrees of q and q′ are constant and their product is equal to the constant degree of q′ ∘q. Thus, it remains to note that under the finite flat surjection q′ : D′′ → D′, the k-degree of the q′-pullback of a line bundle ℒ′ on D′ (respectively of a 0-dimensional closed subscheme Z′ of D′) is deg(q′) times the k-degree of ℒ′ on D′ (respectively the k-degree of Z′). ■

It is now well-posed to define

deg_{k}(ℒ) := ∑_{C} deg_{k}(ℒ |_{C}), deg_{k}(Z) = ∑_{C} deg_{k}(Z ∩ C).

(We do not define deg_{k}(ℒ) via cohomological methods on D because in positive characteristic the coherent cohomology of D may be nonzero in arbitrarily high degree.) The formation of these degrees is easily checked to be unaffected by any extension of the ground field, even though the formation of the set of connected components of D generally does not commute with such base change.

Lemma B.2.2.4. If Z is the schematic image of z ∈ D(k) then deg_{k}(Z) = 1/# Aut(z).

Proof. We may and do replace D with its unique connected component containing Z, so D is connected. Choose a finite flat surjection q : D′ → D from a scheme D′. We may and do assume D′ is connected. The finite map z : Spec(k) → D has schematic image Z that is reduced, so generic flatness is applicable to the finite surjection z : Spec(k) → Z. Openness on the source for the flat locus of any fppf morphism implies that z : Spec(k) → Z is a finite flat covering, necessarily with a constant degree d > 0 since Z is connected. Thus, the projections Aut(z) = z ×_{D} z = z ×_{Z} z ⇉ Spec(k) are finite flat with degree d, so our problem is to show that deg_{k}(Z) = 1/d.

By definition, deg_{k}(Z) = deg_{k}(q^{−1}(Z))/ deg(q). Since q^{−1}(Z) → Z is finite flat with constant degree deg(q), the projection pr_{2} : q^{−1}(Z) ×_{Z,z} Spec(k) → Spec(k) is finite flat with constant degree deg(q). But q^{−1}(Z) ×_{Z,z} Spec(k) is a base change of z and hence is finite flat with constant degree d over the finite k-scheme q^{−1}(Z) via pr_{1}. Thus, d| deg(q) and q^{−1}(Z) has k-degree deg(q)/d. We conclude that deg_{k}(q^{−1}(Z))/ deg(q) = 1/d. ■

The following variant will be useful when working on the Deligne–Mumford stack X_{0}(N)_{Q}.

Lemma B.2.2.5. If D is Deligne–Mumford then for any effective Cartier divisor Z on D the degree deg_{k}(Z) is equal to deg_{k}(𝒪_{D}(Z)), where 𝒪_{D}(Z) denotes the inverse of the invertible coherent ideal of Z in 𝒪_{D}.

Proof. We may assume D is connected and choose a connected finite flat scheme cover q : D′ → D, so q has constant fiber degree d. Since q^{∗}(𝒪_{D}(Z)) ≃ 𝒪_{D′}(q^{−1}(Z)), multiplying both sides by d reduces the problem to its well-known analogue on the scheme D′. ■

Now consider a proper flat Artin stack X with pure relative dimension 1 over R such that there exists a finite flat scheme cover h : X′ → X; we postpone the regularity hypotheses on X and X′ until after we introduce several more definitions.

Since the map h_{0} : X′_{0} → X_{0} between special fiber is a finite flat scheme cover of X_{0}, the preceding considerations are applicable to X_{0} as well as to any closed substack of X_{0} with pure dimension 1. In particular, if Z_{0} is a nowhere-dense closed substack of X_{0} and if ℒ_{0} is an invertible sheaf on X_{0} then we may define deg_{k}(Z_{0}) and deg_{k}(ℒ_{0}) via the preceding formalism applied to X_{0} over k (e.g., using connected components of X′_{0} covering connected components of X_{0}); the same holds for the generic fiber X_{F} over the field F. We make two further definitions that imitate the case of arithmetic surfaces:

Definition B.2.2.6. For any invertible sheaf ℒ on X and effective Cartier divisor D ⊂ X that is a closed substack of X_{0}, we define

(B.2.2.1)  ℒ .D := deg_{k}(ℒ |_{D}).

For effective Cartier divisors D_{1}, D_{2} ⊂ X with D_{2} a closed substack of X_{0}, we define

D_{1}.D_{2} := 𝒪_{X}(D_{1}).D_{2} = deg_{k}(𝒪_{X}(D_{1})|_{D_{2}}).

The preceding definitions are invariant under a local extension R → R′ of discrete valuation rings. (Note that X_{R′} may not be regular even if X is regular.) The next result uses regularity to establish familiar properties.

Proposition B.2.2.7. Assume X is regular and that it admits a regular finite flat scheme cover h : X′ → X. The pairing ℒ .D is additive in ℒ, and if D = D′ + D′′ for effective Cartier divisors D′ and D′′ on X (necessarily closed substacks of X_{0}) then ℒ .D = ℒ .D′ + ℒ .D′′. Moreover, deg_{k}(ℒ_{0}) = deg_{F} (ℒ_{F}) for any invertible sheaf ℒ on X.

For effective Cartier divisors D_{1}, D_{2} ⊂ X that are closed substacks of X_{0}, the pairing D_{1}.D_{2} is symmetric and if they share no irreducible component in their supports then D_{1}.D_{2} = deg_{k}(D_{1} ∩ D_{2}).

Proof. We may extend scalars to make R strictly henselian without harming the regularity hypotheses, so we may and do assume k is separably closed. It also suffices to work separately over the distinct connected components of X and X′, so we may and do assume X and X′ are connected. In particular, h has constant fiber degree.

We claim that X′_{0} is connected (so X_{0} is also connected). Since X′ is a connected normal scheme, it is also integral. Thus, the finite flat R-algebra 𝒪(X′) is a normal domain [EGA, II, 8.8.6.1] and so it is Dedekind. By R-finiteness, 𝒪(X′) must be semi-local. Thus, the henselian property of R forces 𝒪(X′) to be a direct product of discrete valuation rings, so 𝒪(X′) is a discrete valuation ring since it is a domain. The Stein factorization f : X′ → Spec 𝒪(X′) has (geometrically) connected fibers and the residue field of 𝒪(X′) is purely inseparable over k (since k = k_{s}), so the special fiber X′_{0} of X′ → Spec R inherits connectedness from that of the special fiber of f.

The finite flat h_{0} : X′_{0} → X_{0} has constant fiber degree equal to that of h. Thus, by the connectedness of X_{0} and X′_{0}, the definitions over k using h_{0} yield formulas

ℒ .D = \frac{h^{∗}(ℒ).h^{−1}(D)}{deg(h)} , D_{1}.D_{2} = \frac{h^{−1}(D_{1}).h^{−1}(D_{2})}{deg(h)}, deg_{k}(D_{1} ∩ D_{2}) = \frac{h^{−1}(D_{1}).h^{−1}(D_{2})}{deg(h)}.

The additivity of ℒ .D in ℒ and the identity ℒ .D = ℒ .D_{1} + ℒ .D_{2} for effective Cartier divisors D_{1}, D_{2} satisfying D_{1} +D_{2} = D therefore follow from the analogue for intersection theory on the connected regular arithmetic surface X′ relative to Spec(R), and likewise for the symmetry of D_{1}.D_{2} for D_{1}, D_{2} ⊆ X_{0} as well as for the equality D_{1}.D_{2} = deg_{k}(D_{1} ∩ D_{2}) when such D_{1} and D_{2} share no common irreducible component.

Finally, we prove the equality deg_{k}(ℒ_{0}) = deg_{F} (ℒ_{F}). We have already seen that X′_{0} and X_{0} are connected, and X′_{F} inherits connectedness from the integral X′ (so X_{F} is connected too). Clearly h_{0} and h_{F} have the same constant fiber degree (namely, that of h), so our problem is reduced to proving the equality of the numbers deg_{k}(h_{0}^{∗}ℒ_{0}) = deg_{k}((h^{∗}ℒ)_{0}) and deg_{F} ((h^{∗}ℒ)_{F}). This equality follows from X′ being proper and flat over R. ■

Now we assume X is regular (as in the preceding proposition) and that it admits a regular finite flat scheme cover h : X′ → X. We have not yet used that X is Deligne–Mumford away from a nowhere-dense closed substack of X_{0}, but this property will now be used.

Lemma B.2.2.8. Every 1-dimensional reduced and irreducible closed substack D ⊂ X is Cartier. For the generic point η of D, h^{−1}(η) consists of finitely many codimension-1 points of X′.

Proof. Let V ⊆ X be the maximal open substack that is Deligne–Mumford, so X − V has preimage in X′ that consists of finitely many closed points in X′_{0}. It follows that D ↦ D ∩ V is a bijective correspondence between the sets of 1-dimensional reduced and irreducible closed substacks of X and of V, with inverse given by “schematic closure”. Using an étale scheme cover of V, we see that the reduced closed substack D ∩ V of V is Cartier. Thus, by computing with a smooth scheme cover of X it follows that the closure D of the Cartier locally closed substack D ∩ V in the regular X is also Cartier.

If η is the generic point of D then its preimage h^{−1}(η) in X′ consists of the generic points of h^{−1}(D) since h is finite flat. But h^{−1}(D) must have all irreducible components of dimension 1 since D is irreducible of dimension 1, so h^{−1}(η) consists of finitely many codimension-1 points of X′. ■

Consider an invertible rational function f on X. By fppf descent from X′, the maximal open substack U ⊂ X on which f is a section of 𝒪_{X}^{×} has closed complement D such that if D ≠ ∅ then the irreducible components D_{i} of D are all 1-dimensional. In particular, if there are no D_{i}’s (i.e., U = X) then f is a global unit on X. Each D_{i} has a unique generic point η_{i}. Since η_{i} lies in the Deligne–Mumford locus of X, there is a well-defined strictly henselian local ring 𝒪_{X,η_{i}} at η_{i}, and this is a discrete valuation ring. Thus, we can define ord_{η_{i}}(f) ∈ Z, and for any closed substack D ⊂ X with irreducible components of dimension 1 we define the generic multiplicity ord_{η}(D) of D at a generic point η of D to be the length of the 0-dimensional strictly henselian local ring of D at the codimension-1 point η of X (which lies in the open Deligne–Mumford locus of D). For example, if η is a generic point of X_{0} = X mod π then the generic multiplicity of D is ord_{η}(π) for a uniformizer π of R.

For any effective Cartier divisor D in X with generic points η_{i} we define the associated Weil divisor

[D] := ∑_{i} ord_{η_{i}}(D_{i})[D_{i}]

in the free abelian group of Weil divisors of X (with [C] denoting the Weil divisor associated to a reduced and irreducible closed substack C ⊂ X of dimension 1). We also define the Weil divisor

div(f) = ∑_{i} ord_{η_{i}}(f)[D_{i}]

for invertible rational functions f on X. It is clear that div(f_{1}f_{2}) = div(f_{1}) + div(f_{2}), and if div(f) = 0 then f is a global unit on X.

For each irreducible component C_{i} of X_{0} equipped with the reduced structure (so C_{i} is Cartier in X, by Lemma B.2.2.8) and for each invertible sheaf ℒ on X, we use (B.2.2.1) to define

ℒ .[C_{i}] := ℒ .C_{i} = deg_{k}(ℒ |_{C_{i}});

this is additive in ℒ and we extend it by additivity in the second variable to define ℒ .Δ for any Weil divisor Δ of X supported in X_{0}. (We do not require Δ to have its multiplicities along the components of X_{0} in its support to be bounded above by those of [X_{0}].)

Proposition B.2.2.7 applied to an effective Cartier divisor D on X that is a closed substack of X_{0} shows that ℒ .D = ℒ .D′ + ℒ .D′′ when D = D′ + D′′ for effective Cartier divisors D′ and D′′ on X, so (by passage to the C_{i}’s) we see that ℒ .D = ℒ .[D] for such D. If Δ_{1} and Δ_{2} are Weil divisors on X with Δ_{2} supported in X_{0} then we may define Δ_{1}.Δ_{2} to be biadditive in such Δ_{1} and Δ_{2} and to coincide with 𝒪_{X}(Δ_{1}).Δ_{2} when Δ_{2} ≤ [X_{0}]; this recovers Definition B.2.2.6 for effective Cartier divisors that are closed substacks of X_{0} when Δ_{2} = [D_{2}] for a closed substack D_{2} of X_{0} that is Cartier in X. In particular,

div(f).Δ = 0

for any invertible rational function f on X and any Weil divisor Δ on X supported in X_{0}.

Corollary B.2.2.9. Choose Weil divisors Δ_{1} and Δ_{2} on X supported inside X_{0}. The pairing Δ_{1}.Δ_{2} is symmetric. If the supports of Δ_{1} and Δ_{2} do not share an irreducible component in common and Δ_{i} is the Weil divisor associated to an effective Cartier divisor D_{i} in X that is a closed substack of X_{0} then

(B.2.2.2)  Δ_{1}.Δ_{2} = deg_{k}(D_{1} ∩ D_{2}).

If {C_{1}, …, C_{n}} is the set of (reduced) irreducible components of X_{0} then the symmetric intersection matrix (C_{i}.C_{j}) is negative semi-definite, and its defect space is 1-dimensional if X → Spec(R) is its own Stein factorization.

Proof. Biadditivity reduces the symmetry of Δ_{1}.Δ_{2} to the case when each is an irreducible component of X_{0} (equipped with the reduced structure). These correspond to closed substacks of X_{0}, so Proposition B.2.2.7 provides the symmetry. The identity (B.2.2.2) also follows from Proposition B.2.2.7.

It remains to prove that (C_{i}.C_{j}) is negative semi-definite, with 1-dimensional defect space when X → Spec(R) is its own Stein factorization. Negative semi-definiteness means Δ.Δ ≤ 0 for all Weil divisors Δ generated by the [C_{i}]’s. To prove such an inequality we may assume R is strictly henselian and pass to the case when X is connected (so X_{0} is also connected). Thus, we can choose a connected regular finite flat scheme cover h : X′ → X and X′_{0} is also connected. The map h has constant degree, and the known intersection theory on X′ relative to Spec(R) (see Proposition B.2.2.2) gives that

Δ.Δ = \frac{h^{−1}(Δ).h^{−1}(Δ)}{deg(h)} ≤ 0

with equality if and only if h^{−1}(Δ) is a Q-multiple of div_{X′}(π) = h^{−1}(div_{X}(π)). Since h-pullback is injective on Weil divisors, it follows that Δ.Δ = 0 if and only if Δ is a Q-multiple of div_{X}(π). ■

The final topic we address in this section is the definition of the Weil divisor div_{ℒ} (s) for an invertible sheaf ℒ on X and a global section s ∈ ℒ (X) that is non-zero on every connected component of X. For this purpose, we temporarily pass to the setting of an invertible sheaf 𝒩 on an arbitrary Artin stack Y, and a global section t ∈ 𝒩 (Y) that is nowhere a zero-divisor in 𝒩 . By computing on a smooth scheme cover of Y we see that the coherent annihilator ideal ℐ_{Y,𝒩 ,s} = Ann_{𝒪_{Y}} (𝒩 /(s)) is invertible on Y. In the setting of our regular X with equipped (ℒ , s), we define the effective Weil divisor

div_{ℒ} (s) = [ℐ_{X,ℒ ,s}^{−1} ].

Clearly h^{−1}(div_{ℒ} (s)) = div_{h^{∗}(ℒ)} (h^{∗}(s)) as effective Weil divisors on X′. Also, by descent from a (necessarily normal) smooth scheme cover of X, the inclusion s : 𝒪_{X} ↪ ℒ uniquely extends to an isomorphism of invertible sheaves 𝒪_{X}(div_{ℒ} (s)) ≃ ℒ.

Proposition B.2.2.10. Let C be an irreducible component of X_{0}. Equipping C with the reduced structure,

deg_{k}(ℒ |_{C}) = div_{ℒ} (s).[C].

In this proposition, the left side is defined using that C is a proper Artin stack over k admitting a finite flat scheme cover.

Proof. Since ℒ ≃ 𝒪_{X}(Δ) for the Weil divisor Δ = div_{ℒ} (s), it suffices to prove more generally that deg_{k}(𝒪_{X}(Δ)|_{C}) = Δ.[C] for any Weil divisor Δ on X. The case of principal Δ is clear, since both sides vanish. In general both sides are additive in Δ, so we may assume that Δ is a reduced and irreducible Weil divisor. Let D be the effective Cartier divisor in X satisfying [D] = Δ (see Lemma B.2.2.8).

The case Δ = [C] is reduced to the case Δ ≠ [C] due to the established principal case applied to Δ = div(π) (whose support contains [C]). Thus, we may assume Δ ≠ [C], so Δ.[C] = deg_{k}(D ∩ C). Hence, it suffices to show that if C is a closed substack of X_{0} that is Cartier in X and if D is an effective Cartier divisor on X then the invertible sheaf 𝒪_{X}(D)|_{C} on C has k-degree deg_{k}(D ∩ C). Passing to strictly henselian R and connected X (hence connected X_{0}), we may use h-pullback and h_{0}-pullback for a connected regular finite flat scheme cover h : X′ → X to reduce the problem to its known analogue on the regular arithmetic surface X′. ■

Since X is Deligne–Mumford away from a 0-dimensional closed substack of X_{0}, by Lemma B.2.2.8 it is now harmless to identify the notions of effective Cartier divisor and effective Weil divisor in X. Thus, for the remainder of this appendix we shall abuse terminology and notation by identifying the concepts of effective Weil divisor and effective Cartier divisor on X without comment. In particular, for an effective Cartier divisor D in X we now write “D” even where we should write “[D]”.

Let us now return to the original motivating problem that is intrinsic to the regular proper flat Artin stack X_{p} := X_{0}(N)_{Z_{(p)}} over Z_{(p)} and its open substack U_{Z_{(p)}} for U as defined after (B.1.2.1). For a nonzero section f ∈ H^{0}(X_{Q}, ω^{⊗k}) such that the Weil divisor div_{ω^{⊗k}}(f) on X_{p} has multiplicity ≥ 0 along the irreducible component of the special fiber containing U mod p, we will show that its multipicity on any other component is bounded below by −e_{p} for an explicit integer e_{p} ≥ 0 depending only on k, p, and N; then p^{e_{p}}f has all multiplicities ≥ 0 and so lies in H^{0}(X_{p}, ω^{⊗k}) due to the normality of X_{p}.

To formulate our task in a more geometric manner, we introduce some notation. Let {C_{1}, …, C_{n}} be the set of (reduced) irreducible components of X mod p, and write

div_{𝒪_{X}} (p) = ∑ μ_{j}C_{j}

(with μ_{j} ≥ 1 the generic multiplicity of C_{j} on X). For an arbitrary nonzero f ∈ H^{0}(X_{Q}, ω^{⊗k}), in the Weil divisor group of X_{p} we may write

div_{ω^{⊗k}}(f) = D + ∑ ν_{j}C_{j}

for integers ν_{j} and an effective “horizontal” divisor D on X_{p}. Replacing f with p^{e}f for e ∈ Z replaces ν_{j} with ν_{j} + eμ_{j}, so each μ_{i}ν_{j} − μ_{j}ν_{i} is unaffected by p-power scaling of f.

In Theorem B.3.1.3 we will give explicit formulas for the μ_{i}’s depending only on p^{ord_{p}(N)}. Thus, it will suffice to show that the absolute differences |μ_{i}ν_{j} − μ_{j}ν_{i}| are bounded solely in terms of k, p, and ord_{p}(N), as then a lower bound ν_{i_{0}} ≥ 0 for some i_{0} implies a negative lower bound on every ν_{j} depending only on k, p, and ord_{p}(N), as required. The invariance of μ_{i}ν_{j} − μ_{j}ν_{i} under p-power scaling of f allows us to restrict attention to nonzero f ∈ H^{0}(X_{p}, ω^{⊗k}) (rather than f ∈ H^{0}(X_{Q}, ω^{⊗k})) for the purpose of bounding the absolute differences |μ_{i}ν_{j} − μ_{j}ν_{i}|.

In §B.2.3 we will bound every |μ_{i_{0}}ν_{j_{0}} − μ_{j_{0}}ν_{i_{0}}| in terms of: deg_{X_{Q}}(ω^{⊗k}) = k deg_{X_{Q}}(ω), the intersection multiplicities C_{i}.C_{j}, and the multiplicities μ_{i}. These parameters all admit explicit formulas in terms of k, p, and N, as we shall see in §B.3. From this we will obtain explicit bounds as in Theorem B.1.3.1 that depend on N only through ord_{p}(N).

B.2.3. Geometric calculations. We work in the same general setup as in §B.2.2 after Lemma B.2.2.5, but we also assume X_{F} is smooth and geometrically connected (so the special fiber X_{0} is geometrically connected). In particular, there is a regular finite flat scheme cover X′ → X and the stack X is Deligne–Mumford away from a closed substack of dimension 0 in X_{0}. We assume that the special fiber X_{0} is reducible.

Let {C_{1}, …, C_{n}} be the set of (reduced) irreducible components of X_{0} (so n ≥ 2); each C_{i} is Cartier in X by Lemma B.2.2.8. Let μ_{j} be the generic multiplicity of C_{j}, which is to say

div_{𝒪_{X}} (π) = ∑ μ_{j}C_{j}.

Let ℒ be a line bundle on X such that deg_{k}(ℒ |_{C_{j}}) ≥ 0 for all j, and let s ∈ ℒ (X) be a nonzero global section, so

div_{ℒ} (s) = D + ∑ ν_{j}C_{j}

for integers ν_{j} ≥ 0 and an effective horizontal divisor D. In this general setting, we claim that the absolute differences |μ_{i_{0}}ν_{j_{0}} −μ_{j_{0}}μ_{i_{0}}| are all bounded by a universal formula in terms of: deg_{F} (ℒ_{F}), the intersection numbers C_{i}.C_{j}, and the multiplicites μ_{i}. Since

(B.2.3.1)  μ_{i}ν_{j} − μ_{j}ν_{i} = \frac{μ_{i}}{μ_{1}} (μ_{1}ν_{j} − μ_{j}ν_{1}) − \frac{μ_{j}}{μ_{1}} (μ_{1}ν_{i} − μ_{i}ν_{1}),

it suffices to restrict attention to bounding |μ_{1}ν_{j} − μ_{j}ν_{1}| for each j ≠ 1.

Since s is a nonzero section of ℒ, so ℒ ≃ 𝒪_{X}(div_{ℒ} (s)), we have

deg_{k}(ℒ |_{C_{j}}) = div_{ℒ} (s).C_{j} = D.C_{j} + ∑_{i} ν_{i}(C_{i}.C_{j}).

Let M denote the symmetric negative semi-definite intersection matrix (C_{i}.C_{j}) and define

ν⃗ := (ν_{1}, …, ν_{n}),

so intersection theory on X ensures that M : Q^{n} → Q^{n} is negative semi-definite with kernel equal to the line spanned by μ⃗ = (μ_{1}, …, μ_{n}) (∑ μ_{j}C_{j} = div_{𝒪_{X}} (π) is principal). Thus, the image hyperplane of M is

H = {a⃗ ∈ Q^{n} | ∑ μ_{i}a_{i} = 0}

since for any b⃗ ∈ Q^{n} and a_{i} := ∑_{j}(C_{i}.C_{j})b_{j} we have

∑_{i} μ_{i}a_{i} = ∑_{j} b_{j} ∑_{i} μ_{i}(C_{i}.C_{j}) = ∑_{j} b_{j}(div(π).C_{j}) = 0.

Remark B.2.3.1. For ν⃗ = (ν_{1}, …, ν_{n}) as above and

a⃗ := M(ν⃗)

we have a_{j} = deg_{k}(ℒ |_{C_{j}}) − D.C_{j}. The identity ∑_{i} μ_{i}a_{i} = 0 above also follows from the pair of identities

deg_{F} (div_{ℒ_{F}} (s_{F})) = deg_{F} ℒ_{F} = deg_{k} ℒ_{0} = ∑ μ_{j} deg_{k}(ℒ |_{C_{j}})

and

deg_{F} (div_{ℒ_{F}} (s_{F})) = deg_{F} (D_{F}) = deg_{k}(D_{0}) = D.div(π) = ∑ μ_{j}(D.C_{j}).

Since deg_{k}(ℒ |_{C_{j}}) ≥ 0 for all j, we have |μ_{i}a_{i}| ≤ deg_{F} ℒ_{F} for all i because μ_{i}a_{i} = μ_{i} deg_{k}(ℒ |_{C_{i}})−μ_{i}(D.C_{j}) is a difference between non-negative integers that are each bounded above by deg_{F} ℒ_{F}.

Rather generally, for any a⃗ such that ∑ μ_{i}a_{i} = 0, the b⃗ ∈ Q^{n} such that M(b⃗) = a⃗ are determined up to adding a Q-multiple of μ⃗, so the differences μ_{i}b_{j} − μ_{j}b_{i} are independent of the choice of such b⃗. Our aim is to give a formula for these differences entirely in terms of a⃗, μ⃗, and the intersection numbers C_{i′}.C_{i′′}. We will focus on the case i = 1 (so j > 1), as this is sufficient by (B.2.3.1).

Let W ⊂ Q^{n} be the subspace of vectors with vanishing initial coordinate, so the natural projection pr : Q^{n} → W away from the initial coordinate restricts to an isomorphism on the hyperplane H (since μ_{1} ≠ 0). Since the hyperplane H is the image of M : Q^{n} → Q^{n}, and the subspace W is complementary to the line ker M (since μ_{1} ≠ 0), the restriction of M to W is an isomorphism W ≃ H. Thus, we can define a composite linear automorphism

T : W ≃^{M|_{W}} H ≃^{pr} W.

Via the evident isomorphism W ≃ Q^{n−1}, the matrix for T is the lower-right (n − 1) × (n − 1) block in the symmetric intersection matrix defining M. The desired formula for the differences μ_{1}b_{j} − μ_{j}b_{1} (j ≥ 2) is given by:

Proposition B.2.3.2. For any a⃗ ∈ H and b⃗ ∈ Q^{n} such that M(b⃗) = a⃗, the point T^{−1}(pr(a⃗)) ∈ W has jth coordinate (μ_{1}b_{j} − μ_{j}b_{1})/μ_{1} for 2 ≤ j ≤ n; that is,

b⃗ − \frac{b_{1}}{μ_{1}} · μ⃗ = T^{−1}(pr(a⃗)).

Proof. Since a⃗ = M(b⃗), in view of the definition of T we just need to observe that the vector b⃗ − (b_{1}/μ_{1})μ⃗ lies in W and so it is the unique point where W meets the congruence class of b⃗ modulo the kernel Qμ⃗ of M. ■

By setting b⃗ = ν⃗ and a_{j} = deg_{k}(ℒ |_{C_{j}}) − D.C_{j} for all j ≥ 2, Proposition B.2.3.2 gives a formula for (μ_{1}ν_{j} − μ_{j}ν_{1})/μ_{1} in terms of the intersection numbers C_{i}.C_{i′} and the numbers a_{2}, …, a_{n}. To make this formula explicit, we have to invert the (n−1)×(n−1) matrix for the linear automorphism T of W = Q^{n−1}.

Example B.2.3.3. Suppose n = 2. In this case,

\frac{μ_{1}ν_{2} − μ_{2}ν_{1}}{μ_{1}} = \frac{a_{2}}{C_{2}.C_{2}} = \frac{μ_{2}a_{2}}{(μ_{2}C_{2}).C_{2}} = −\frac{μ_{2}a_{2}}{μ_{1}(C_{1}.C_{2})}

(since the Weil divisor μ_{1}C_{1} +μ_{2}C_{2} = div_{𝒪_{X}} (π) has vanishing intersection pairing against any Weil divisor supported in X_{0}), so

μ_{1}ν_{2} − μ_{2}ν_{1} = −\frac{μ_{2}a_{2}}{C_{1}.C_{2}}

where a_{2} = deg_{k}(ℒ |_{C_{2}}) − D.C_{2}. By Remark B.2.3.1 we have |μ_{2}a_{2}| ≤ deg_{F} (ℒ_{F}) and hence

(B.2.3.2)  |μ_{1}ν_{2} − μ_{2}ν_{1}| ≤ \frac{deg_{F} (ℒ_{F})}{C_{1}.C_{2}} .

Remark B.2.3.4. Assume in the setting of Example B.2.3.3 that μ_{1} = μ_{2} = 1, so we are using the upper bound deg_{F} (ℒ_{F}) on deg_{k}(ℒ |_{C_{j}}). For example, with X = X_{0}(p) and C_{1} the component generically corresponding to the “multiplicative” level structure (so C_{1} ≃ X(1)), this amounts to replacing deg(ω^{⊗k}|_{C_{1}}) = k/24 with the upper bound deg(ω^{⊗k}) = k(p + 1)/24.

Thus, in this special case the use of (B.2.3.2) does not recover as good a bound as is obtained by Deligne and Rapoport in [DR, VII, 3.19–3.20], where the exact value deg(ω^{⊗k}|_{C_{1}}) = k/24 is used. However, we can modify the computation in such cases (in a manner that is specific to n = 2) so that we recover the bound of Deligne and Rapoport. This requires the explicit determination of certain intersection pairings in §B.3.1, so we postpone it until Theorem B.3.2.1.

Example B.2.3.5. Suppose n = 3. We identify W with Q^{2} via pr_{23}, so the matrix for T : W ≃ W is

[T] = (\begin{matrix}C_{2}.C_{2} & C_{2}.C_{3} \\ C_{2}.C_{3} & C_{3}.C_{3} \end{matrix}) .

Setting a_{j} = deg_{k}(ℒ |_{C_{j}}) − D.C_{j} for j = 2, 3 (so |a_{j}| ≤ deg_{F} (ℒ_{F})/μ_{j}, as in Example B.2.3.3), we have

μ_{1}^{−1} · (\begin{matrix}μ_{1}ν_{2} − μ_{2}ν_{1} \\ μ_{1}ν_{3} − μ_{3}ν_{1} \end{matrix}) = (\begin{matrix}C_{2}.C_{2} & C_{2}.C_{3} \\ C_{2}.C_{3} & C_{3}.C_{3} \end{matrix})^{−1} (\begin{matrix}a_{2} \\ a_{3} \end{matrix}) .

This yields upper bounds on |μ_{1}ν_{j} − μ_{j}ν_{1}| in terms of the intersection numbers C_{i}.C_{i′}, μ⃗, and deg_{F} (ℒ_{F}). Note that

C_{2}.C_{2} = μ_{2}^{−1}(μ_{1}(C_{1}.C_{2}) + μ_{3}(C_{3}.C_{2})), C_{3}.C_{3} = μ_{3}^{−1}(μ_{1}(C_{1}.C_{3}) + μ_{2}(C_{2}.C_{3})).

B.3. Explicit formulas.

B.3.1. Numerology for X_{0}(N). The preceding considerations are applicable to the stack X_{0}(N) (N ≥ 1) that is Deligne–Mumford away from the cuspidal substack in characteristics p for which p^{2}|N [Co, 3.2.7]. In particular, this stack is Deligne–Mumford around supersingular geometric points in any positive characteristic.

Consider the line bundle ℒ = ω^{⊗k} on X = X_{0}(N), where ω = ω_{Γ_{0}(N)} is the invertible pushforward of the relative dualizing sheaf of the universal generalized elliptic curve over X_{0}(N). The degree deg_{F_{p}}(ℒ_{F_{p}}) is equal to deg_{Q}(ℒ_{Q}) = k deg_{Q}(ω_{Q}) for any prime p, so to compute the degree of ℒ on the fibers of X_{0}(N) over Spec(Z) we just need to compute deg_{Q}(ω_{Q}).

The forgetful map of stacks X_{0}(N) → X(1) is proper, quasi-finite, and flat, but even over Q it is not finite (more specifically, not relatively representable in schemes) when N is not square-free. The problem is that if we choose d|N such that gcd(d, N/d) > 1 then for the standard d-gon E_{d} over \overline{Q} and the cyclic subgroup G of order N generated by (ζ, 1 mod d) ∈ G_{m} × Z/dZ with a primitive Nth root of unity ζ, the non-trivial group of automorphisms μ_{d} ∩ μ_{N/d} of (E_{d}, G) induces the identity on the contraction c(E_{d}) ≃ E_{1} away from the identity component. This implies that the fiber of X_{0}(N) → X(1) over ∞ : Spec(Q) → X(1) is not a scheme.

Nonetheless, the degree of ω behaves as if X_{0}(N) → X(1) is finite flat:

Lemma B.3.1.1. For any N ≥ 1, deg_{Q}(ω_{Γ_{0}(N)}) = [Γ(1) : Γ_{0}(N)] deg_{Q}(ω_{Γ(1)}).

Proof. The forgetful map X_{1}(N) → X(1) is finite flat by [Co, Thm. 4.1.1(1)], and the map X_{1}(N) → X_{0}(N) is finite flat by [Co, Lemma 4.2.3]. These maps are defined moduli-theoretically via contraction operations that have no effect on the relative identity component of the smooth locus of a generalized elliptic curve, and by computations over Z[1/N] away from cusps we see that their constant fiber degrees are [Γ(1) : Γ_{1}(N)] and [Γ_{0}(N) : Γ_{1}(N)] respectively.

For any generalized elliptic curve f : E → S, f_{∗}(ω_{E/S}) is naturally identified with e^{∗}(ω_{E/S}). Thus, ω_{Γ_{1}(N)} on X_{1}(N) is identified with the pullbacks of both ω_{Γ_{0}(N)} and ω_{Γ(1)} under the forgetful maps X_{1}(N) ⇉ X_{0}(N), X(1). Hence,

[Γ_{0}(N) : Γ_{1}(N)] deg_{Q}(ω_{Γ_{0}(N)}) = deg_{Q}(ω_{Γ_{1}(N)}) = [Γ(1) : Γ_{1}(N)] deg_{Q}(ω_{Γ(1)}).

Dividing throughout by [Γ_{0}(N) : Γ_{1}(N)] yields the result. ■

The line bundle ω_{Γ(1)}^{⊗12} over X(1)_{Q} admits a generating section away from ∞, namely Ramanujan’s cusp form Δ, and div_{ω_{Γ(1)}^{⊗12}}(Δ) = ∞ by q-expansion considerations. But deg_{Q}(∞) = 1/2 due to Lemma B.2.2.4

(since the standard 1-gon has automorphism group Z/(2)), so the isomorphism ω_{Γ(1)}^{⊗12} ≃ 𝒪_{X(1)}(∞) implies that 12 deg_{Q}(ω_{Γ(1)}) = deg_{Q}(∞) = 1/2. Hence:

Corollary B.3.1.2. For all k, N ≥ 1, deg_{Q}(ω_{Γ_{0}(N)}^{⊗k}) = k[Γ(1) : Γ_{0}(N)]/24.

Pick a prime p|N and let m = ord_{p}(N) ≥ 1, so N = p^{m}N′ with p ∤ N′. The geometry of X_{0}(N) mod p is worked out in [KM, Ch. 13] except that the language of stacks is not used there: prime-to-p level structure is adjoined to create a “rigid” moduli problem represented by a scheme, and cusps are not treated moduli-theoretically. By using the argeement between the moduli-theoretic approach in [Co] (which does incorporate the cusps) and the methods in [KM] for cases with “enough” level structure (such as Γ_{1}(Nqℓ) for distinct primes q, ℓ ≥ 5 not dividing N), we get most of:

Theorem B.3.1.3. The stack X_{0}(N)_{F_{p}} has m + 1 irreducible components {C_{a}}_{0≤a≤m}. Equipping each C_{a} with the reduced structure, each is smooth and geometrically irreducible with

deg_{F_{p}}(ω|_{C_{a}}) > 0

for all a.

Any two such distinct components meet at precisely the supersingular points, and at any supersingular point ξ of X_{0}(N)_{\overline{F}_{p}} the completed local ring is isomorphic to

\overline{F}_{p}[[u, v]]/((u − v^{p^{m}})(v − u^{p^{m}}) ∏_{1≤a≤m−1}(u^{p^{a}} − v^{p^{m−a}})^{p−1})

with its quotient 𝒪̂_{C_{a},ξ} defined by the radical of the equation u^{p^{a}} = v^{p^{m−a}}. In particular, the generic multiplicity of C_{a} is given by μ_{a} = p^{min(a,m−a)} (e.g., μ_{0}, μ_{m} = 1) and for 0 ≤ a < b ≤ m the 0-dimensional intersection stack C_{a} ∩ C_{b} is reduced if a ≤ m/2 ≤ b whereas otherwise each of its points has multiplicity

m_{a,b} = \begin{cases}p^{m−2b}, & a < b < m/2 \\ p^{2a−m}, & m/2 < a < b. \end{cases}

Proof. This is [KM, Thm. 13.4.7] except for the inequalities deg_{F_{p}}(ω|_{C_{a}}) > 0 for all a. To prove this positivity we may instead work with ω^{⊗12}, which is the pullback of 𝒪(∞) on X(1) along the quasifinite flat proper forgetful map X_{0}(N)_{F_{p}} → X(1)_{F_{p}}. This latter map is not finite (nor even relatively representable in schemes) when p^{2}|N, and the same holds for each C_{a} → X(1)_{F_{p}}. Nonetheless, the effect on degree under such pullbacks does behave as if it is finite flat, exactly as in Lemma B.3.1.1ff., so ω|_{C_{a}} has positive F_{p}-degree for every a. ■

Corollary B.3.1.4. For any two distinct irreducible components C and C′ of X_{0}(N) mod p,

C.C′ ≥ [Γ(1) : Γ_{0}(N′)] · ∑_{ξ} \frac{1}{# Aut(ξ)} = [Γ(1) : Γ_{0}(N′)] · \frac{p − 1}{24},

where the sum is over all isomorphism classes of supersingular elliptic curves over \overline{F}_{p}.

Proof. Since intersection numbers can be computed over \overline{F}_{p}, we may and do work with 𝒳 := X_{0}(N)_{\overline{F}_{p}}. By Theorem B.3.1.3, 𝒞 := C_{\overline{F}_{p}} and 𝒞′ := C′_{\overline{F}_{p}} are irreducible components of 𝒳 with the reduced structure

and 𝒞 ∩ 𝒞′ is a closed substack whose support consists of precisely the supersingular \overline{F}_{p}-points with each such point having a common multiplicity (depending on C and C′).

Any supersingular elliptic curve E over \overline{F}_{p} admits a unique Drinfeld cyclic subgroup scheme of order p^{m}, namely the kernel of the m-fold relative Frobenius morphism (for which a Drinfeld generator is given by 0). Thus, if N = p^{m} (i.e., N′ = 1) then Lemma B.2.2.4 and Proposition B.2.2.7 give that

𝒞.𝒞′ ≥ ∑_{ξ} \frac{1}{# Aut(ξ)} = \frac{p − 1}{24}

(with equality precisely when C and C′ meet transversally). More generally, consider the map h : X_{0}(N) → X_{0}(p^{m}) that “forgets” the N′-part of the level structure. This generically étale proper map generally is not relatively representable near cusps but is finite etale of constant degree deg(h) := [Γ_{0}(p^{m}) : Γ_{0}(N)] = [Γ(1) : Γ_{0}(N′)] away from the cusps. Also, the pullback operation Y ⇝ h^{−1}(Y) defines a bijection between the sets of irreducible components of X_{0}(p^{m})_{F_{p}} and X_{0}(N)_{F_{p}}.

Let D and D′ be the irreducible components of X_{0}(p^{m})_{F_{p}} that satisfy C = h^{−1}(D) and C′ = h^{−1}(D′), so by using a finite flat scheme cover of X_{0}(N) that is also finite flat over X_{0}(p^{m}) (such as by adjoining suitable auxiliary level structure) we see that C.C′ = deg(h)D.D′ = [Γ(1) : Γ_{0}(N′)]D.D′. Since D.D′ ≥ (p−1)/24, we are done. ■

Corollary B.3.1.5. Choose a prime p|N. For sufficiently large even k, the p-torsion subgroup of the finite abelian group M_{k}(Γ_{0}(N), Z)/M_{k,Z} is non-trivial.

Since M_{k}(Γ_{0}(N), C) = 0 for odd k, the parity condition on k cannot be removed. In the special case k = 4 and N = p = 5, f in §5.1 represents a non-trivial class.

Proof. By flat base change considerations, it suffices to show that M_{k}(Γ, R)/M_{k,R} ≠ 0 for the strict henselization R = Z_{(p)}^{sh}. Let κ be the residue field of R, and let D be the reduced effective Cartier divisor on the regular stack X_{R} such that the support of D is the unique irreducible component of the special fiber X_{κ} passing through ∞. Finally, let [x_{0}] be the 0-dimensional closed substack of X_{κ} that is the schematic image of a non-cuspidal non-CM point x_{0} : Spec κ → X_{κ} away from D and let j : [x_{0}] → X_{R} be the canonical closed immersion.

The non-cuspidality ensures that [x_{0}] is Deligne–Mumford, so the map x_{0} : Spec κ → [x_{0}] is finite since it is a proper quasi-finite map from a scheme to a Deligne–Mumford stack. By the structure of DM-stacky points, x_{0} identifies [x_{0}] with the quotient stack [Spec(κ)/G_{0}] for the finite constant group G_{0} = Spec(κ) ×_{x_{0},X_{κ},x_{0}} Spec(κ) that is the automorphism group of the object over κ classified by x_{0}. Since x_{0} is non-cuspidal and non-CM, clearly G_{0} = Z/2Z. Thus, Pic([x_{0}]) consists of isomorphism classes of 1-dimensional κ-vector spaces L equipped with an action by G_{0} = Z/2Z, so the G_{0}-action on L^{⊗2} is trivial. Hence, j^{∗}(ω_{R})^{⊗k} ≃ 𝒪_{[x_{0}]} for any even k.

Choose a prime ℓ ≥ 5 not dividing Np, and let p^{a} denote the p-part of the degree of the finite flat map q : X_{1}(Nℓ) → X_{0}(N) = X with X_{1}(Nℓ) a scheme over Z[1/ℓ] (and hence over Z_{(p)}). For any coherent sheaf 𝒢 on X_{R} we claim that if k is sufficiently large (without a parity constraint) then H^{1}(X_{R}, ω^{⊗k} ⊗𝒢) is killed by p^{a}. (There is no finite flat scheme cover of X = X_{0}(N)_{κ} with degree prime to p when ord_{p}(N) > 1, due to degree considerations at the cusps in characteristic p whose automorphism scheme has order divisible by p.) Granting this, consider the twist of 𝒪_{X_{R}} ↠ j_{∗}(𝒪_{[x_{0}]}) by the invertible ideal sheaf 𝒪_{X_{R}}(−(a + 1)D) (which is trivial near [x_{0}]); this twist is a surjection φ : 𝒪_{X_{R}}(−(a + 1)D) → j_{∗}(𝒪_{[x_{0}]}). Let ℐ denote its coherent kernel, so ω_{R}^{⊗k} ⊗ ℐ is the kernel of the natural surjection ω_{R}^{⊗k}(−(a + 1)D) → j_{∗}(j^{∗}(ω_{R}^{⊗k})) obtained by twisting φ by ω_{R}^{⊗k}. For even k the line bundle j^{∗}(ω_{R}^{⊗k}) on [x_{0}] is trivial, and the obstruction to lifting a global section of this line bundle to a global section f of ω_{R}^{⊗k}(−(a+1)D) lies in the cohomology group H^{1}(X_{R}, ω_{R}^{⊗k} ⊗ ℐ) that is killed by p^{a}. Thus, there exists f ∈ M_{k,R} generating p^{a}ω_{R}^{⊗k} near [x_{0}] and vanishing to order at least a + 1 along X_{κ} near ∞_{κ}. Clearly f has q-expansion that vanishes modulo p^{a+1}, so (1/p^{a+1})f represents a nontrivial class in M_{k}(Γ, R)/M_{k,R}.

It remains to show that for any coherent sheaf 𝒢 on X_{R}, H^{1}(X_{R}, ω_{R}^{⊗k} ⊗ 𝒢) is killed by p^{a} for all sufficiently large k (without a parity constraint). Since q : X_{1}(Nℓ) → X_{0}(N) is a finite flat scheme cover whose degree has p-part p^{a}, by a trace argument it suffices to prove that for any coherent sheaf ℱ on X_{1}(N)_{R} (such as q^{∗}𝒢) if k is sufficiently large then H^{1}(X_{1}(Nℓ)_{R}, q^{∗}(ω)_{R}^{⊗k} ⊗ ℱ) = 0. By the standard base change theorems for coherent cohomology on schemes, it suffices to prove the analogous vanishing on the special fiber X_{κ}, so it is enough to show that the line bundle q^{∗}(ω)_{κ} on X_{1}(Nℓ)_{κ} is ample.

As C varies through the irreducible components of the stacky curve X_{0}(N)_{κ} (giving C the reduced structure), the pullbacks q^{−1}(C) vary through the irreducible components of X_{1}(Nℓ)_{κ} (equipped with possibly non-reduced scheme structure). For each such C the rational number deg_{κ}(ω_{κ}|_{C}) is positive by Theorem B.3.1.3, so q^{∗}(ω)_{κ} has positive degree on the irreducible 1-dimensional proper κ-scheme q^{−1}(C). Hence, q^{∗}(ω)_{κ}|_{q^{−1}(C)} is ample for all such C, so q^{∗}(ω)_{κ} is ample as desired. ■

The interested reader will easily check that Corollary B.3.1.5 remains true with the same proof when using cusp forms in place of modular forms: one just has to introduce a twist by the invertible ideal sheaf defining the Cartier closed substack of cusps in some of the sheaf-theoretic calculations.

B.3.2. Multiplicity bounds for Γ_{0}(N). Consider a prime p|N and let m := ord_{p}(N), so N = p^{m}N′ with p ∤ N′. We seek an e = e(p, k, N) so that p^{e}M_{k}(Γ_{0}(N), Z_{(p)}) ⊆ H^{0}(X_{0}(N)_{Z_{(p)}}, ω^{⊗k}).

Theorem B.3.2.1. We can choose e(p, k, N) to depend on N only through m := ord_{p}(N) > 0. For m = 1, 2, 3 it can be chosen to be ⌈kp^{m−1}(m + 1/(p − 1))⌉.

Finding an explicit formula for e(p, k, N) via our methods requires understanding the inverse of an m × m matrix of intersection numbers, so it is not clear if ⌈kp^{m−1}(m + 1/(p − 1))⌉ suffices for m > 3.

Proof. The case m = 1 requires special care (relative to arguments that work for m > 1), so we first assume m = 1 and adapt the argument in [DR, VII, 3.19–3.20] to show that |ν_{2} − ν_{1}| ≤ kp/(p − 1) (regardless of the prime-to-p part of N).

Consider Example B.2.3.3 with C_{1} equal to the component X_{0}(N′)_{F_{p}} of X_{0}(N)_{F_{p}} corresponding to multiplicative p-part in the level structure (away from the supersingular locus). This yields the equality

(ν_{2} − ν_{1})(C_{1}.C_{2}) = D.C_{1} − deg_{F_{p}}(ω^{⊗k}|_{C_{1}}) = D.C_{1} − k[Γ(1) : Γ_{0}(N′)]/24

(using Lemma B.3.1.1 for Γ_{0}(N′) over the characteristic-p point of Spec(Z[1/N′])).

The key trick is to observe that

D.C_{1} = (1/2)D.(C_{1} + C_{2}) + (1/2)D.(C_{1} − C_{2}),

and that the Weil divisor C_{1} + C_{2} = div_{X_{0}(N)}(p) has vanishing pairing against any vertical divisor in the mod-p fiber. It follows that D.(C_{1} + C_{2}) is unaffected by replacing D with the divisor D + ν_{1}C_{1} + ν_{2}C_{2} of an invertible rational section of ω_{Γ_{0}(N)}^{⊗k}, so

D.(C_{1} + C_{2}) = deg_{F_{p}}(ω_{Γ_{0}(N)}^{⊗k}) = deg_{Q}(ω_{Γ_{0}(N)}^{⊗k}) = k[Γ(1) : Γ_{0}(N)]/24.

Since N = pN′ with p ∤ N′, we have

[Γ(1) : Γ_{0}(N)] = [Γ(1) : Γ_{0}(N′)][Γ(1) : Γ_{0}(p)] = (p + 1)[Γ(1) : Γ_{0}(N′)].

Hence,

(ν_{2} − ν_{1})(C_{1}.C_{2}) = k[Γ(1) : Γ_{0}(N′)](−1/24 + (1/2)((p + 1)/24)) + (1/2)D.(C_{1} − C_{2})
= (k/2)[Γ(1) : Γ_{0}(N′)](p − 1)/24 + (1/2)D.(C_{1} − C_{2}).

The horizontality of D implies that |D.(C_{1} − C_{2})| ≤ D.(C_{1} + C_{2}), and by Corollary B.3.1.4

C_{1}.C_{2} ≥ [Γ(1) : Γ_{0}(N′)](p − 1)/24.

By cancelling [Γ(1) : Γ_{0}(N′)](p − 1)/24 throughout, we conclude that

|ν_{2} − ν_{1} − k/2| ≤ (k/2)(p + 1)/(p − 1) = (k/2)(1 + 2/(p − 1)),

so |ν_{2} − ν_{1}| ≤ k + k/(p − 1) = kp/(p − 1), as desired.

Now suppose m = 2, so for the three irreducible components C_{0}, C_{1}, C_{2} of X_{0}(N) mod p the respective multiplicities are μ_{0} = μ_{2} = 1 and μ_{1} = p − 1. Example B.2.3.5 gives that

(\begin{matrix}ν_{1} − (p − 1)ν_{0} \\ ν_{2} − ν_{0} \end{matrix}) = (\begin{matrix}C_{1}.C_{1} & C_{1}.C_{2} \\ C_{1}.C_{2} & C_{2}.C_{2} \end{matrix})^{−1} (\begin{matrix}a_{1} \\ a_{2} \end{matrix}) ,

and

|μ_{j}a_{j}| ≤ deg_{Q}(ω^{⊗k}) = kp(p + 1)[Γ(1) : Γ_{0}(N′)]/24

by Remark B.2.3.1 and Corollary B.3.1.2. The description of the completed local rings at geometric supersingular points in Theorem B.3.1.3 gives (via Lemma B.2.2.4 with m = 2) that C_{i} meets C_{j} transversally whenever i ≠ j, so

C_{0}.C_{2} = C_{1}.C_{2} = C_{0}.C_{1} = ∑_{(E,G)} \frac{1}{# Aut(E, G)} =: I

where the sum is taken over all \overline{F}_{p}-isomorphism classes of pairs consisting of a supersingular elliptic curve E equipped with a subgroup scheme G of order N that is cyclic in the Drinfeld sense. Also, div(p) = C_{0} + (p − 1)C_{1} + C_{2}, so

(p − 1)C_{1}.C_{1} = −(C_{0}.C_{1} + C_{2}.C_{1}) = −2I, C_{2}.C_{2} = −(C_{0}.C_{2} + (p − 1)C_{1}.C_{2}) = −pI,

so

(B.3.2.1)  (\begin{matrix}C_{1}.C_{1} & C_{1}.C_{2} \\ C_{1}.C_{2} & C_{2}.C_{2} \end{matrix})^{−1} = I^{−1} (\begin{matrix}−2/(p − 1) & 1 \\ 1 & −p \end{matrix})^{−1} = \frac{p − 1}{I(p + 1)} (\begin{matrix}−p & −1 \\ −1 & −2/(p − 1) \end{matrix}) .

By Example B.2.3.5 we conclude that

|ν_{1} − (p − 1)ν_{0}| = \frac{(p − 1)(pa_{1} + a_{2})}{I(p + 1)}, |ν_{2} − ν_{0}| = \frac{(p − 1)}{I(p + 1)}(a_{1} + 2a_{2}/(p − 1)).

Moreover, I ≥ [Γ(1) : Γ_{0}(N′)](p − 1)/24 by Corollary B.3.1.4, and our upper bound on |μ_{j}a_{j}| says

|a_{2}|, (p − 1)|a_{1}| ≤ kp(p + 1)[Γ(1) : Γ_{0}(N′)]/24.

Thus, after cancellation of [Γ(1) : Γ_{0}(N′)]/24 we arrive at

|ν_{1} − (p − 1)ν_{0}| ≤ \frac{2p − 1}{p − 1}kp, |ν_{2} − ν_{0}| ≤ \frac{3}{p − 1}kp.

It follows that if ν_{0} ≥ 0 then

ν_{1}, ν_{2} ≥ −\frac{2p − 1}{p − 1}kp = − (2 + \frac{1}{p − 1}) kp.

Hence, when ord_{p}(N) = 2 we can take e(p, k, N) to be ⌈(2 + 1/(p − 1))kp⌉.

Finally, we adapt the treatment of the case m = 2 to handle general m > 1. Let C_{0}, …, C_{m} be as in Theorem B.3.1.3, and let T be the m × m matrix whose ij-entry (1 ≤ i, j ≤ m) is C_{i}.C_{j}. If 0 ≤ i < j ≤ m then C_{i}.C_{j} = C_{j}.C_{i} = m_{i,j}I where

I := ∑_{(E,G)} \frac{1}{# Aut(E, G)} ≥ [Γ(1) : Γ_{0}(N′)] · \frac{p − 1}{24}

and m_{i,j} = 1 except when i < j < m/2 or m/2 < i < j, in which case m_{i,j} = p^{m−2j} or m_{i,j} = p^{2i−m} respectively. For ease of notation, we define m_{j,i} = m_{i,j} when j > i.

For i ≥ 1, the self-intersection C_{i}.C_{i} satisfies

μ_{i}C_{i}.C_{i} = − ∑_{h≠i}(μ_{h}C_{h}.C_{i}) = −I · ∑_{h≠i} m_{h,i}μ_{h}

(where the sums include h = 0, with μ_{0} = 1). In other words, C_{i}.C_{i} = −I · ∑_{h≠i} m_{h,i}μ_{h}/μ_{i}, so T^{−1} = (1/I)B^{−1} where the matrix B = (b_{ij}) with 1 ≤ i, j ≤ m has entries b_{ij} = m_{i,j} for i ≠ j and b_{ii} = − ∑_{h≠i} m_{h,i}μ_{h}/μ_{i} (including h = 0). Each b_{ij} depends only on p^{m} (not on the prime-to-p part N′ of N) because the same holds for the multiplicities μ_{i} of the C_{i} in X_{0}(N)_{F_{p}} by Theorem B.3.1.3. (Explicitly, μ_{i} = p^{min(i,m−i)} for 0 ≤ i ≤ m.)

Define ν⃗ = (ν_{1}, …, ν_{m}) and μ⃗ = (μ_{1}, …, μ_{m}) in Q^{m}, and let Δ be the diagonal m × m matrix whose ith diagonal entry is μ_{i} (1 ≤ i ≤ m). Since μ_{0} = 1, applying Proposition B.2.3.2 (to C_{0}, …, C_{m}) gives that

(B.3.2.2)  ν⃗ − ν_{0} · μ⃗ = (1/I)(b_{ij})^{−1}a⃗

where |μ_{j}a_{j}| ≤ kp^{m−1}(p + 1)[Γ(1) : Γ_{0}(N′)]/24. Let w⃗ ∈ Q^{m} be the vector whose jth entry is w_{j} := μ_{j}a_{j}. Clearly

|w_{j}/I| = |μ_{j}a_{j}/I| ≤ kp^{m−1}(p + 1)/(p − 1),

so inserting the trivial identity a⃗ = Δ^{−1}w⃗ into (B.3.2.2) gives that

|ν_{i} − μ_{i}ν_{0}| ≤ kp^{m−1}(p + 1)/(p − 1) ∑_{j} |c^{ij}|

where (c^{ij}) is the inverse of the matrix

(c_{ij}) := Δ · B

with c_{ij} = μ_{i}m_{i,j} if j ≠ i and c_{ii} = − ∑_{h≠i} m_{h,i}μ_{h} (including h = 0). Thus, if ν_{0} ≥ 0 then

ν_{i} ≥ −\frac{kp^{m−1}(p + 1)}{p − 1} · ∑_{j} |c^{ij}|,

so e(p, k, N) may be chosen to be any integer at least

(B.3.2.3)  kp^{m−1} · \frac{p + 1}{p − 1} max_{i} ∑_{j} |c^{ij}|.

This lower bound visibly depends on N only through its p-part p^{m}.

By explicit computation, if m = 3 then

(c^{ij}) = \frac{−1}{(p + 1)^{3}} (\begin{matrix}(p + 1)^{2} & p(p + 1) & p(p + 1) \\ p(p + 1) & 2p^{2} + p + 1 & 2p^{2} \\ p + 1 & 2p & 3p + 1 \end{matrix}) ,

so all c^{ij} are negative, as also holds for m = 2 via inspection of (B.3.2.1). (Perhaps c^{ij} < 0 for all m, but we have no proof.) The sufficient lower bound in (B.3.2.3) on choices for e(p, k, N) when ord_{p}(N) = 3 is now easily checked to be bounded above by kp^{2}(3 + 1/(p − 1)). ■

References

[BDP1] Bertolini, M., Darmon, H., and Prasanna, K., Generalised Heegner cycles and p-adic Rankin L-series, Duke Math Journal, 162, No 6, (2013) 1033–1148.

[BDP2] Bertolini, M., Darmon, H., and Prasanna, K., p-adic Rankin L-series and rational points on CM elliptic curves, Pacific Journal of Math., Vol. 260 (2012), No. 2, 261–303.

[BDP3] Bertolini, M., Darmon, H., and Prasanna, K., Chow-Heegner points on CM elliptic curves and values of p-adic L-functions, IMRN, first published online November 1, 2012 doi:10.1093/imrn/rns237.

[Be] Beĭlinson, A. A., Height pairing between algebraic cycles. K-theory, arithmetic and geometry (Moscow, 19841986), 1-25, Lecture Notes in Math., 1289, Springer, Berlin, 1987.

[Bl-1] Bloch, Spencer, Algebraic cycles and values of L-functions. J. Reine Angew. Math. 350 (1984), 94-108. [Bl-2] Bloch, Spencer, Algebraic cycles and values of L-functions. II. Duke Math. J. 52 (1985), no. 2, 379-397.

[BK] Bloch, Spencer; Kato, Kazuya, L-functions and Tamagawa numbers of motives. The Grothendieck Festschrift, Vol. I, 333-400, Progr. Math., 86, Birkhuser Boston, Boston, MA, 1990.

[Cer] Ceresa, G., C is not algebraically equivalent to C^{−} in its Jacobian. Ann. of Math. (2) 117 (1983), no. 2, 285-291.

[Ch] T. Chinburg, “Minimal models for curves over Dedekind rings” (Ch. XIII) in Arithmetic Geometry (Cornell/Silverman ed.), Springer–Verlag, 1986.

[Cl] Clemens, Herbert, Homological equivalence, modulo algebraic equivalence, is not finitely generated. Inst. Hautes Études Sci. Publ. Math. No. 58 (1983), 19-38 (1984).

[Co] B. Conrad, Arithmetic moduli of generalized elliptic curves. J. Inst. Math. Jussieu 6 (2007), no. 2, 209–278.

[DR] Deligne, Pierre; & Rapoport, Michael, Les schémas de modules de courbes elliptiques. (French) Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), pp. 143–316. Lecture Notes in Math., Vol. 349, Springer, Berlin, 1973.

[dS] de Shalit, Ehud, Iwasawa theory of elliptic curves with complex multiplication, Perspectives in Mathematics 3, Academic Press Inc., Boston, MA, 1987.

[E] Edixhoven, S.J., Comparison of integral structures on spaces of modular forms of weight two, and computation of spaces of forms mod 2 of weight one, Journal of the Math. Institute of Jussieu 5 (2006), no. 1, 1–34.

[GS] Goldstein, Catherine; Schappacher, Norbert, Séries d’Eisenstein et fonctions L de courbes elliptiques à multiplication complexe. (French) [Eisenstein series and L functions of elliptic curves with complex multiplication] J. Reine Angew. Math. 327 (1981), 184-218.

[Gre] Greenberg, Ralph, On the critical values of Hecke L-functions for imaginary quadratic fields. Invent. Math. 79 (1985), no. 1, 79-94.

[Gr-1,2] Griffiths, Phillip A., On the periods of certain rational integrals. I, II. Ann. of Math. (2) 90 (1969), 460-495; ibid.

(2) 90 1969 496-541.

[EGA] A. Grothendieck, Élements de Géométrie Algébrique, Math. Publ. IHES 4, 8, 11, 17, 20, 24, 28, 32, 1960–7.

[Har] Harris, Bruno, Homological versus algebraic equivalence in a Jacobian. Proc. Nat. Acad. Sci. U.S.A. 80 (1983), no. 4 i., 1157-1158.

[Jan] Jannsen, Uwe, Equivalence relations on algebraic cycles.The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), 225-260, NATO Sci. Ser. C Math. Phys. Sci., 548, Kluwer Acad. Publ., Dordrecht, 2000.

[Katz1] Katz, Nicholas M., p-adic interpolation of real analytic Eisenstein series Ann. of Math., 104, no. 3 (1976), pp. 459-571.

[Katz2] Katz, Nicholas M., p-adic L-functions for CM fields. Invent. Math. 49 (1978), 199-297.

[KM] Katz, Nicholas M.; Mazur, Barry, Arithmetic moduli of elliptic curves, Annals of Mathematics Studies 108, Princeton Univ. Press, 1985.

[KMe] Katz, Nicholas M.; Messing, William, Some consequences of the Riemann hypothesis for varieties over finite fields. Invent. Math. 23 (1974), 73-77.

[Kis] Kisin, Mark, Potential semi-stability of p-adic étale cohomology. Israel J. Math. 129 (2002), 157-173.

[Mat] H. Matsumura, Commutative ring theory, Translated from the Japanese by M. Reid. Second edition. Cambridge Studies in Advanced Mathematics, 8. Cambridge University Press, Cambridge, 1989.

[Nek1] Nekovár̆, Jan, p-adic Abel-Jacobi maps and p-adic heights. The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), 367-379, CRM Proc. Lecture Notes, 24, Amer. Math. Soc., Providence, RI, 2000.

[Nek2] Nekovár̆, Jan, On p-adic height pairings. S’eminaire de Th’eorie des Nombres, Paris, 1990-91, 127-202, Progr. Math., 108, Birkhäuser Boston, Boston, MA, 1993.

[Ol] Olsson, Martin, On proper coverings of Artin stacks, Advances in Mathematics 198 (2005), 93–106.

[PS] Peters, Chris A. M.; Steenbrink, Joseph H., M., Mixed Hodge structures., Springer-Verlag, Berlin, 2008.

[Schap] Schapppacher, Norbert; Periods of Hecke characters Lecture Notes in Mathematics 1301, Springer-Verlag, Berlin, 1988.

[Schoen1] Schoen, Chad, Complex multiplication cycles on elliptic modular threefolds. Duke Math. J. 53 (1986), no. 3, 771-794.

[Schoen2] Schoen, Chad, Complex multiplication cycles and a conjecture of Belinson and Bloch. Trans. Amer. Math. Soc. 339 (1993), no. 1, 87-115.

[Scholl] Scholl, A. J., Motives for modular forms, Invent. Math. 100 (1990), no. 2, 419-430.

[Shim] Shimura, Goro, On some arithmetic properties of modular forms of one and several variables. Ann. of Math. (2) 102 (1975), no. 3, 491–515.

[Wa] Waldspurger, Jean-Loup. Sur les valeurs de certaines fonctions L automorphes en leur centre de symétrie. (French) [Values of certain automorphic L-functions at their center of symmetry] Compositio Math. 54 (1985), no. 2, 173–242.

[Zh] Zhang, Shouwu, Heights of Heegner cycles and derivatives of L-series. Invent. Math. 130 (1997), no. 1, 99-152.