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

MASSIMO BERTOLINI HENRI DARMON KARTIK PRASANNA

Abstract: We outline a new construction of rational points on CM elliptic curves using cycles on higher dimensional varieties, contingent on certain cases of the Tate conjecture. This construction admits complex and p-adic analogs that are defined independently of the Tate conjecture. In the p-adic case, we show unconditionally that the points so constructed are in fact rational using p-adic Rankin L-functions and a p-adic Gross-Zagier type formula proved in our previous articles [BDP-gz] and [BDP-cm]. In the complex case, we are unable to prove rationality (or even algebraicity) but we can verify it numerically in several cases.

Contents

1. Introduction 1 2. Motives and Chow-Heegner points 6 2.1. Motives for rational and homological equivalence 7 2.2. The motive of a Hecke character 8 2.3. Deligne-Scholl motives 9 2.4. Modular parametrisations attached to CM forms 10 2.5. Generalised Heegner cycles and Chow-Heegner points 14 2.6. A special case 16 3. Chow-Heegner points over ℂ_{p} 16 3.1. The p-adic Abel-Jacobi map 16 3.2. Rationality of Chow-Heegner points over ℂ_{p} 18 4. Chow-Heegner points over ℂ 21 4.1. The complex Abel-Jacobi map 21 4.2. Numerical experiments 23 References 26

1. Introduction

The theory of Heegner points supplies one of the most fruitful approaches to the Birch and Swinnerton-Dyer conjecture, leading to the best results for elliptic curves of analytic rank one. In spite of attempts to broaden the scope of the Heegner point construction ([BDG], [DL], [Da], [Tr],...), all provable, systematic constructions of algebraic points on elliptic curves still rely on parametrisations of elliptic curves by modular or Shimura curves. The primary goal of this article is to explore new constructions of rational points on elliptic curves and abelian varieties in which, loosely speaking, Heegner divisors are replaced by higher-dimensional algebraic cycles on certain modular varieties. In general, the algebraicity of the resulting points depends on the validity of ostensibly difficult cases of the Hodge or Tate conjectures. One of the main theorems of this article (Theorem 4 of the Introduction) illustrates how these algebraicity statements can sometimes be obtained unconditionally by exploiting the connection between the relevant “generalised Heegner cycles” and values of certain p-adic Rankin L-series.

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

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

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

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

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

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

(1.2)  Φ : J_{0}(N)⟶A

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

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

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

which follows by combining the Gross-Zagier formula with a method of Kolyvagin (cf. [Gr]). The theory of Heegner points is also the key ingredient in the proof of the main results in [BDP-cm].

Given a variety X (defined over ℚ, say), let CH^{j}(X)(F) denote the Chow group of codimension j algebraic cycles on X defined over a field F modulo rational equivalence, and let CH^{j}(X)_{0}(F) denote the subgroup of null-homologous cycles. Write CH^{j}(X) and CH^{j}(X)_{0} for the corresponding functors on ℚ-algebras. Via the natural equivalence CH^{1}(X_{0}(N))_{0} = J_{0}(N), the map Φ of (1.2) can be recast as a natural transformation

(1.3)  Φ : CH^{1}(X_{0}(N))_{0}⟶A.

It is tempting to generalise (1.3) by replacing X_{0}(N) by a variety X over ℚ of dimension d > 1, and CH^{1}(X_{0}(N))_{0} by CH^{j}(X)_{0} for some 0 ≤ j ≤ d. Any element Π of the Chow group CH^{d+1−j}(X × A)(ℚ) induces a natural transformation

(1.4)  Φ : CH^{j}(X)_{0}⟶A

sending Δ ∈ CH^{j}(X)_{0}(F) to

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

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

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

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

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

ψ_{A}((a)) = ε_{K}(a mod \sqrt{−D})a.

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

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

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

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

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

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

Our construction of Chow-Heegner points is based on the following conjecture which is formulated (for more general K, without the class number one hypothesis) in Section 2.

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

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

for some element c_{ψ,K} in K^{×}, where

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

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

Remark 2. In fact, in the special case considered above, using that A is defined over ℚ and not just K, one can arrange the cycle Π^{?} to be defined over ℚ (if it exists at all !). Then c_{ψ,K} lies in ℚ^{×} and by appropriately scaling Π^{?} we may further arrange that c_{ψ,K} = 1. This would simplify some of the discussion below, see for example the commutative diagram (1.13). However we have chosen to retain the constant c_{ψ,K} in the rest of the introduction in order to give the reader a better picture of the more general situation considered in the main text where the class number of K is not 1 and the curve A can only be defined over some extension of K.

Remark 3. The rationale for Conjecture 1 is explained in Section 2.4, where it is shown to follow from the Tate conjecture on algebraic cycles. To the authors’ knowledge, the existence of Π^{?} is known only in the following cases:

(1) r = 0, where it follows from Faltings’ proof of the Tate conjecture for a product of curves over number fields;

(2) (r, D) = (1, −4) (see Remark 2.4.1 of [Scha]) and (1, −7), where it can be proved using the theory of Shioda-Inose structures and the fact that W_{r} is a singular K3 surface ([El]);

(3) r = 2 and D = −3, (Schoen - see [Scho2], Sec. 1). For general values of r and D, Conjecture 1 appears to lie rather deep and might be touted as a good “proving ground” for the validity of the Hodge and Tate conjectures. One of the main results of this paper–Theorem 4 below–uses p-adic methods to establish unconditionally a consequence of Conjecture 1, leading to the construction of rational points on A. The complex calculations of the last section likewise lend numerical support for an (ostensibly deeper) complex analogue of Theorem 4. Sections 3 and 4 may therefore be viewed as providing indirect support (of a theoretical and experimental nature, respectively) for the validity of Conjecture 1.

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

X_{r} := W_{r} × A^{r}.

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

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

as in (1.4), that is defined over K, i.e., there is a natural map

Φ_{F}^{?} : CH^{r+1}(X_{r})_{0}(F)⟶A(F) ⊗ ℚ

for any field F containing K. Furthermore, it satisfies the equation

(1.7)  Φ_{dR}^{?∗}(ω_{A}) = c_{ψ,K} · ω_{θ_{ψ}} ∧ η_{A}^{r}.

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

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

where [λ] denotes the element of End_{K}(A) corresponding to λ. (See Proposition 2.11 for details.)

The article [BDP-gz] introduced and studied a collection of null-homologous, r-dimensional algebraic cycles on X_{r}, i.e., elements of the source CH^{r+1}(X_{r})_{0,ℚ} of the map (1.6), referred to as generalised Heegner cycles. These cycles, whose precise definition is recalled in Section 2.5, extend the notion of Heegner cycles on Kuga-Sato varieties considered in [Scho1], [Ne1] and [Zh]. They are indexed by isogenies φ : A⟶A′, and are defined over abelian extensions of K. It can be shown that they generate a subspace of CH^{r+1}(X_{r})_{0,ℚ}(K^{ab}) of infinite dimension. The map Φ_{K^{ab}}^{?} (if it exists) transforms these generalised Heegner cycles into points of A(K^{ab}) ⊗ ℚ. It is natural to expect that the resulting collection {Φ_{K^{ab}}^{?}(Δ_{φ})}_{φ:A⟶A′} of Chow-Heegner points generates an infinite dimensional subspace of A(K^{ab}) ⊗ ℚ, and that it gives rise to an “Euler system” in the sense of Kolyvagin.

In the classical situation where r = 0, the variety X_{r} is just a modular curve and (as mentioned above) the existence of Φ^{?} follows from Faltings’ proof of the Tate conjecture for products of curves. When r ≥ 1, Section 3 uses p-adic methods to show that an alternate cohomological construction of Φ_{K^{ab}}^{?}(Δ_{φ}) gives rise in many cases to algebraic points on A with the expected field of rationality and offers, therefore, some theoretical evidence for the existence of Φ^{?}. We now describe this construction briefly.

Let p be a rational prime split in K and fix a prime 𝔭 of K above p. As explained in Remark 2.12 of Section 2.4, even in the absence of knowing the Tate conjecture, one can still define a natural G_{K} := Gal(K̄/K)-equivariant projection

(1.9)  Φ_{et,𝔭}^{∗} : H_{et}^{2r+1}(\overline{X}_{r}, ℚ_{p})(r+1)⟶H_{et}^{1}(\overline{A}, ℚ_{p})(1) = V_{p}(A),

where V_{p}(A) is the p-adic Galois representation arising from the p-adic Tate module of A. A priori, this last map is only well-defined up to an element in ℚ_{p}^{×}. We normalise it by by embedding K in ℚ_{p} via 𝔭 and requiring that the map

Φ_{dR,𝔭}^{∗} : H_{dR}^{2r+1}(X_{r}/ℚ_{p})⟶H_{dR}^{1}(A/ℚ_{p})

obtained by applying to Φ_{et,𝔭}^{∗} the comparison functor between p-adic étale cohomology and deRham cohomology over p-adic fields satisfies

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

where ω_{θ_{ψ}} and ω_{A} are as in Conjecture 1, and η_{A} is defined in (1.8). We can then define the following p-adic avatars of Φ^{?} without invoking the Tate conjecture:

(a) The map Φ_{F}^{et}:

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

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

where H^{1}(F, M) denotes the continuous Galois cohomology of a G_{F} := Gal(F̄/F)-module M. The maps (1.11) and (1.9) can be combined to give a map

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

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

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

where

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

is the projective limit of the connecting homomorphisms arising in the p^{n}-descent exact sequences of Kummer theory, and c_{ψ,K} is the element in K^{×} from Conjecture 1 viewed as living in ℚ_{p}^{×} via the embedding of K in ℚ_{p} corresponding to 𝔭.

(b) The map Φ_{F}^{(v)}: When F is a number field, (1.13) suggests that the image of Φ_{F}^{et} is contained in the Selmer group of A over F, and this can indeed be shown to be the case. In fact, one can show that for every finite place v of F, the image of Φ_{F_{v}}^{et} is contained in the images of the local connecting homomorphisms

δ_{v} : A(F_{v}) ⊗ ℚ⟶H^{1}(F_{v}, V_{p}(A)).

In particular, fixing a place v of F and replacing F by its v-adic completion F_{v}, we can define a map Φ_{F}^{(v)} by the commutativity of the following local counterpart of the diagram (1.13):

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

As will be explained in greater detail in Section 3, when v is a place lying over 𝔭, the map Φ_{F}^{(v)} can also be defined by p-adic integration, via the comparison theorems between the p-adic étale cohomology and the de Rham cohomology of varieties over p-adic fields.

The main Theorem of this paper, which is proved in Section 3, relates the Selmer classes of the form Φ_{F}^{et}(Δ) when F is a number field and Δ is a generalised Heegner cycle, to global points in A(F). We will only state a special case of the main result, postponing the more general statements to Section 3.2. Assume for Theorem 4 below that the field K has odd discriminant, that the sign in the functional equation for L(ψ_{A}, s) is −1, so that the Hasse-Weil L-series L(A/ℚ, s) = L(ψ_{A}, s) vanishes to odd order at s = 1, and that the integer r is odd. In that case, the theta series θ_{ψ} belongs to the space S_{r+2}(Γ_{0}(D), ε_{K}) of cusp forms on Γ_{0}(D) of weight r + 2 and character ε_{K} := (\frac{·}{D} ). In particular, the variety W_{r} is essentially the rth Kuga-Sato variety over the modular curve X_{0}(D). Furthermore, the L-series L(ψ_{A}^{2r+1}, s) has sign +1 in its functional equation, and L(ψ_{A}^{2r+1}, s) therefore vanishes to even order at the central point s = r + 1.

Theorem 4. Let Δ_{r} be the generalised Heegner cycle in CH^{r+1}(X_{r})_{0,ℚ}(K) attached to the identity isogeny 1 : A⟶A. The cohomology class Φ_{K}^{et}(Δ_{r}) belongs to δ(A(K) ⊗ ℚ). More precisely, there is a point P_{D} ∈ A(K) ⊗ ℚ (depending on D but not on r) such that

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

where m_{D,r} ∈ ℤ satisfies

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

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

L′(ψ_{A}, 1) ≠ 0.

This result is proved, in a more general form, in Theorems 3.3 and 3.5 of Section 3.2. For an even more general (but less precise) statement in which the simplifying assumptions imposed in Theorem 4 are considerably relaxed, see Theorem 3.4.

Remark 5. When L(A, s) has a simple zero at s = 1, it is known a priori that the Selmer group Sel_{p}(A/K) is of rank one over K ⊗ ℚ_{p}, and agrees with δ(A(K) ⊗ ℚ_{p}). It follows directly that

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

The first part of Theorem 4 is significantly stronger in that it involves the rational vector space A(K) ⊗ ℚ rather than its p-adification. This stronger statement is not a formal consequence of the one-dimensionality of the Selmer group. Indeed, its proof relies on invoking Theorem 2 of [BDP-cm] after relating the local point Φ_{K_{𝔭}}^{(𝔭)}(Δ) ∈ A(K_{𝔭}) ⊗ ℚ = A(ℚ_{p}) ⊗ ℚ to the special value ℒ_{p}(ψ_{A}^{∗}) of the Katz two-variable p-adic L-function that arises in that theorem.

Finally, we discuss the picture over the complex numbers. Section 4.1 describes a complex homomorphism

Φ_{ℂ} : CH^{r+1}(X_{r})_{0}(ℂ)⟶A(ℂ)

which is defined analytically by integration of differential forms on X_{r}(ℂ), without invoking Conjecture 1, but agrees with Φ_{ℂ}^{?} (up to multiplication by some nonzero element in O_{K}) when the latter exists. This map is defined using the complex Abel-Jacobi map on cycles introduced and studied by Griffiths and Weil, and is the complex analogue of the homomorphism Φ_{K_{𝔭}}^{(𝔭)}. The existence of the global map Φ_{K}^{?} predicted by the Hodge or Tate conjecture would imply the following algebraicity statement:

Conjecture 6. Let H be a subfield of K^{ab} and let Δ_{φ} ∈ CH^{r+1}(X_{r})_{0,ℚ}(H) be a generalised Heegner cycle defined over H. Then (after fixing an an embedding of H into ℂ),

Φ_{ℂ}(Δ_{φ}) belongs to A(H) ⊗ ℚ,

and

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

While ostensibly weaker than Conjecture 1, Conjecture 6 has the virtue of being more readily amenable to experimental verification. Section 4 explains how the images of generalised Heegner cycles under Φ_{ℂ} can be computed numerically to high accuracy, and illustrates, for a few such Δ_{φ}, how the points Φ_{ℂ}(Δ_{φ}) can be recognized as algebraic points defined over the predicted class fields. In particular, extensive numerical verifications of Conjecture 6 are carried out, for fairly large values of r.

On the theoretical side, this conjecture appears to lie deeper than its p-adic counterpart, and we were unable to provide any theoretical evidence for it beyond the fact that it follows from the Hodge or Tate conjectures. It might be argued that calculations of the sort that are performed in Section 4 provide independent numerical confirmation of these conjectures for certain specific Hodge and Tate cycles on the (2r + 2)-dimensional varieties W_{r} × A^{r+1}, for which the corresponding algebraic cycles seem hard to produce unconditionally.

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

Acknowledgements: We are grateful to the anonymous referee whose comments helped us to considerably clarify and improve our exposition.

2. Motives and Chow-Heegner points

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

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

C^{m}(X) := CH^{m}(X) = C^{m}(X)/ ∼ .

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

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

Definition 2.1. A motive over F with coefficients in E is a triple (X, e, m) where X/F is a nonsingular projective variety, e ∈ Corr^{0}(X, X)_{E} is an idempotent, and m is an integer.

Definition 2.2. The category M_{F,E} of Chow motives is the category whose objects are motives over F with coefficients in E, with morphisms defined by

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

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

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

M_{F,E} → M_{F,E}^{hom},

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

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

M_{B} := e · (H^{∗}(X(ℂ), ℚ)(m) ⊗ E).

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

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

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

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

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

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

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

Moreover, there are natural comparison isomorphisms

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

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

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

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

Remark 2.3. If F is a p-adic field, one also has a comparison isomorphism

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

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

Remark 2.4. Our definition of motives with coefficients coincides with Language B of Deligne [Del]. There is an equivalent way of defining motives with coefficients (the Language A) where the objects are motives M in M_{F,ℚ} equipped with the structure of an E-module: E → End(M), and morphisms are those that commute with the E-action. We refer the reader to Sec. 2.1 of loc. cit. for the translation between these points of view.

2.2. The motive of a Hecke character. In this section we recall how to attach a motive to an algebraic Hecke character ψ of an quadratic imaginary field K of infinity type (r, 0). (The reader is referred to Section 2.1 of [BDP-cm] for out notations and conventions regarding algebraic Hecke characters.) This generalises the exposition of Section 2.2 of [BDP-cm], where we recall how an abelian variety with complex multiplication is attached to a Hecke character of K of infinity type (1, 0). For more general algebraic Hecke characters which are not of type (1, 0), one no longer has an associated abelian variety. Nevertheless, such a character still gives rise to a motive over K with coefficients in the field generated by its values.

Suppose that ψ : 𝔸_{K}^{×} → ℂ^{×} is such a Hecke character and let E_{ψ} be the field generated over K by the values of ψ on the finite idèles. Pick a finite Galois extension F of K such that ψ_{F} := ψ ∘ N_{F/K} satisfies the equation

ψ_{F} = ψ_{A}^{r},

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

We construct motives M(ψ_{F}) ∈ M_{F,ℚ}, M(ψ_{F})_{K} ∈ M_{F,K} associated to ψ_{F} by considering an appropriate piece of the middle cohomology of the variety A^{r} over F. Similarly to the Introduction, write [α] for the element of End_{F} (A) ⊗_{ℤ} ℚ corresponding to an element α ∈ K. Define an idempotent e_{r} = e_{r}^{(1)} ∘ e_{r}^{(2)} ∈ Corr^{0}(A^{r}, A^{r})_{ℚ} by setting

e_{r}^{(1)} := (\frac{\sqrt{−D} + [\sqrt{−D}]}{2\sqrt{−D}})^{⊗r} + (\frac{\sqrt{−D} − [\sqrt{−D}]}{2\sqrt{−D}})^{⊗r} , e_{r}^{(2)} := (\frac{1 − [−1]}{2})^{⊗r} .

Let M(ψ_{F}) be the motive in M_{F,ℚ} defined by

M(ψ_{F}) := (A^{r}, e_{r}, 0),

and let M(ψ_{F})_{K} denote the motive in M_{F,K} obtained (in Language A) by making K act on M(ψ_{F}) via its diagonal action on A^{r}. The ℓ-adic étale realisation M(ψ_{F})_{K,ℓ} is free of rank one over K ⊗ ℚ_{ℓ}, and G_{F} acts on it via ψ_{F}, viewed as a (K ⊗ ℚ_{ℓ})^{×}-valued Galois character:

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

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

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

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

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

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

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

It can be shown that after extending coefficients to E_{ψ}, the motive M(ψ_{F})_{K} descends to a motive M(ψ) ∈ M_{K,E_{ψ}}, whose ℓ-adic realisation is a free rank one module over E_{ψ} ⊗ ℚ_{ℓ} on which G_{K} acts via the character ψ. In this article however we shall only make use of the motives M(ψ_{F}) and M(ψ_{F})_{K}.

2.3. Deligne-Scholl motives. Let S_{r+2}(Γ_{0}(N), ε) be the space of cusp forms on Γ_{0}(N) of weight r + 2 and nebentype character ε. In this section, we will let ψ be a Hecke character of K of infinity type (r+1, 0). This Hecke character gives rise to a theta-series

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

as in Proposition 2.10 of [BDP-cm], with N := D · N_{K/ℚ}(𝔣) and ε := ε_{ψ} · ε_{K}, where ε_{ψ} is the central character of ψ (see [BDP-cm], Defn. 2.2) and ε_{K} is the quadratic Dirichlet character associated to the extension K/ℚ. Observe that the subfield E_{θ_{ψ}} of ℚ̄ generated by the Fourier coefficients a_{n}(θ_{ψ}) is always contained in E_{ψ} and if ψ is a self-dual character (see loc. cit. Defn. 3.4) then E_{θ_{ψ}} is a totally real field, and E_{ψ} = E_{θ_{ψ}}K.

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

X^{2} − a_{p}(θ_{ψ})X + ε(p)p^{r+1}.

This representation is realised in the middle ℓ-adic cohomology of a variety which is fibered over a modular curve. More precisely let Γ := Γ_{ε}(N) ⊂ Γ_{0}(N) be the congruence subgroup of SL_{2}(ℤ) attached to f, defined by

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

Writing H for the Poincaré upper half place of complex numbers with strictly positive imaginary part, and H^{∗} for H ∪ P_{1}(ℚ), let C denote the modular curve whose complex points are identified with Γ\H^{∗}. Let W_{r} be the r-th Kuga-Sato variety over C. It is a canonical compactification and desingularisation of the r-fold self-product of the universal elliptic curve over C. (See for example [BDP-gz], Chapter 2 and the Appendix for more details on this definition.)

Remark 2.5. The article [BDP-gz] is written using Γ_{1}(N) level structures. The careful reader may therefore wish to replace Γ = Γ_{ε}(N) by Γ_{1}(N) throughout the rest of the paper, and make the obvious modifications. For example, in the definition of P_{ψ}^{?}(χ) in 2.26 below, one would need to take a trace before summing over Pic(O_{c}). This is explained in more detail in [BDP-co], §4.2.

Theorem 2.6. (Scholl) There is a projector e_{θ_{ψ}} ∈ Corr_{0}^{0}(W_{r}, W_{r}) ⊗ E_{θ_{ψ}} whose associated Grothendieck motive M(θ_{ψ}) := (W_{r}, e_{θ_{ψ}}, 0) satisfies (for all ℓ)

M(θ_{ψ})_{ℓ} ≃ V_{ℓ}(θ_{ψ})

as E_{θ_{ψ}}[G_{ℚ}]-modules.

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

The de Rham realisation

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

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

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

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

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

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

and Γ acts by the rule

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

The holomorphic (r+1)-form

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

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

Fil^{0} M(θ_{ψ})_{dR} = M(θ_{ψ})_{dR},
Fil^{1} M(θ_{ψ})_{dR} = ⋯ = Fil^{r+1} M(θ_{ψ})_{dR} = E_{θ_{ψ}} · ω_{θ_{ψ}},
Fil^{r+2} M(θ_{ψ})_{dR} = 0.

The following proposition compares the Deligne-Scholl motive associated to θ_{ψ} with the CM motives constructed in the previous section in the main case of interest to us. We will suppose that ψ is a self-dual Hecke character of K of infinity type (r + 1, 0), and as in the previous section that F is a finite Galois extension of K such that ψ_{F} = ψ_{A}^{r+1} for some elliptic curve A over F with CM by O_{K}.

Proposition 2.7. For every finite prime ℓ, the ℓ-adic representations associated to the motives M(θ_{ψ})|_{F} and M(ψ_{F}) ⊗_{ℚ} E_{θ_{ψ}} are isomorphic as (E_{θ_{ψ}} ⊗ ℚ_{ℓ})[G_{F}]-modules.

Proof. It suffices to check this after further tensoring with E_{ψ} (over E_{θ_{ψ}}). Note that the ℓ-adic realization of M(θ_{ψ})|_{F} ⊗_{E_{θ_{ψ}}} E_{ψ} is a rank-2 (E_{ψ} ⊗ ℚ_{ℓ})[G_{F}]-module on which G_{F} acts as ψ_{F,ℓ} ⊕ ψ_{F,ℓ}^{∗}, where ψ^{∗} is the Hecke character of K obtained from ψ by composing with complex conjugation on 𝔸_{K}^{×}. On the other hand, since E_{χ} = KE_{θ_{χ}} ≃ K ⊗_{ℚ} E_{θ_{χ}}, the ℓ-adic realization of M(ψ_{F}) ⊗_{ℚ} E_{ψ} is a rank-2 (E_{ψ} ⊗ ℚ_{ℓ})[G_{F}]-module on which G_{F} acts as ψ_{ℓ} ⊕ ψ̄_{ℓ}. However the characters ψ^{∗} and ψ̄ are equal since ψ is self-dual, so the result follows. □

2.4. Modular parametrisations attached to CM forms. In this section, we will explain how the Tate conjectures imply the existence of algebraic cycle classes generalising those in Conjecture 1 of the Introduction. Recall the Chow groups CH^{d}(V)(F) defined in the Introduction.

Conjecture 2.8 (Tate). Let V be a smooth projective variety over a number field F. Then the ℓ-adic étale cycle class map

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

is surjective.

A class in the target of (2.8) is called an ℓ-adic Tate cycle. The Tate conjecture will be used in our constructions through the following simple consequence.

Lemma 2.9. Let V_{1} and V_{2} be smooth projective varieties of dimension d over a number field F, and let e_{j} ∈ Corr^{0}(V_{j}, V_{j}) ⊗ E (for j = 1, 2) be idempotents satisfying

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

Let M_{j} := (V_{j}, e_{j}, 0) be the associated motives over F with coefficients in E, and suppose that the ℓ-adic realisations of M_{1} and M_{2} are isomorphic as (E ⊗ ℚ_{ℓ})[G_{F}]-modules. If Conjecture 2.8 is true for V_{1} × V_{2}, then there exists a correspondence Π ∈ CH^{d}(V_{1} × V_{2})(F) ⊗ E for which

(1) the induced morphism

(2.9)  Π_{ℓ}^{∗} : (M_{1})_{ℓ}⟶(M_{2})_{ℓ}

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

(2) the induced morphism

(2.10)  Π_{dR}^{∗} : (M_{1})_{dR}⟶(M_{2})_{dR}

is an isomorphism of E ⊗ F -vector spaces.

Proof. Let

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

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

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

where the superscript ^{∨} in the first line denotes the E ⊗ ℚ_{ℓ}-linear dual, the second line follows from the Poincaré duality, and the third from the Künneth formula. By Conjecture 2.8, there are elements α_{1}, …, α_{t} ∈ E ⊗ ℚ_{ℓ} and cycles Π_{1}, …, Π_{t} ∈ CH^{d}(V_{1} × V_{2})(F) satisfying

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

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

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

satisfies condition 1 in the statement of Lemma 2.9. Condition 2 is verified by embedding F into one of its ℓ-adic completions F_{λ} and applying Fontaine’s comparison functor to (2.9) in which source and targets are de Rham representations of G_{F_{λ}}. This shows that Π_{dR}^{∗} induces an isomorphism on the de Rham cohomology over F_{λ} ⊗ E, and part 2 follows. □

The following proposition (in which, to ease notations, we identify differential forms with their image in de Rham cohomology) justifies Conjecture 1 of the Introduction. Notations are as in Section 2.2 and 2.3, with ψ a self-dual Hecke character of infinity type (r + 1, 0).

Proposition 2.10. If the Tate conjecture is true for W_{r} × A^{r+1}, then there is an algebraic cycle Π^{?} ∈ CH^{r+1}(W_{r} × A^{r+1})(F) ⊗ E_{θ_{ψ}} such that

(2.11)  Π_{dR}^{?∗}(ω_{A}^{r+1}) = c_{ψ,F} · ω_{θ_{ψ}}.

for some c_{ψ,F} ∈ (F ⊗_{ℚ} E_{θ_{ψ}})^{×}.

Proof. Let M_{1} and M_{2} be the motives M(ψ_{F}) ⊗_{ℚ} E_{θ_{ψ}} and M(θ_{ψ})|_{F} in M_{F,E_{θ_{ψ}}}. By Prop. 2.7 the ℓadic realisations of M_{1} and M_{2} are isomorphic. Part (1) of Lemma 2.9 implies, assuming the validity of Conjecture 2.8, the existence of a correspondence Π^{?} in CH^{r+1}(W_{r} × A^{r+1})(F) ⊗ E_{θ_{ψ}} which induces an isomorphism on the ℓ-adic and de Rham realisations of M_{1} and M_{2}. The isomorphism on de Rham realizations respects the Hodge filtrations and therefore sends the class ω_{A}^{r+1} to a unit F ⊗_{ℚ} E_{θ_{ψ}}-rational multiple of ω_{θ_{ψ}}, hence the proposition follows. □

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

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

Let X_{r} be the F -variety

X_{r} = W_{r} × A^{r}.

After setting

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

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

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

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

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

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

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

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

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

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

Recall that η_{A} ∈ H_{dR}^{1}(A/F) is defined as in (1.8) of the Introduction.

Proposition 2.11. The image of the class ω_{A} ∈ Ω^{1}(A/F) ⊂ H_{dR}^{1}(A/F) under Φ_{dR}^{?} is given by

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

where c_{ψ,F} is as in Prop. 2.10.

Proof. Suppose that

Π^{?} = ∑_{j} m_{j}Z_{j}

is an E_{θ_{ψ}}-linear combination of codimension (r+1) subvarieties of Z. The cycle class map is given by

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

where

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

By Proposition 2.10 and the construction of Π_{dR}^{?}, we have

(2.15)  Π_{dR}^{?∗}(ω_{A}^{r+1}) = c_{ψ,F} · ω_{θ_{ψ}},

and

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

By definition of Π_{dR}^{?}, equation (2.15) can be rewritten as

(2.17)  cl_{Π^{?}}(π_{0}^{∗}(α) ∧ π_{12}^{∗}(ω_{A}^{r+1})) = ⟨α, c_{ψ,F} · ω_{θ_{ψ}}⟩_{W_{r}},  for all α ∈ H_{dR}^{r+1}(W_{r}/F) ⊗ E_{θ_{ψ}},

while (2.16) shows that

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

Equation (2.17) can also be rewritten as

(2.19)  cl_{Φ^{?}}(π_{01}^{∗}(α ∧ ω_{A}^{r}) ∧ π_{2}^{∗}(ω_{A})) = ⟨α ∧ ω_{A}^{r}, c_{ψ,F} · ω_{θ_{ψ}} ∧ η_{A}^{r}⟩_{X_{r}},

while equation (2.18) implies that, for all α ∈ H_{dR}^{r+1}(W_{r}/F) ⊗ E_{θ_{ψ}} and all 1 ≤ j ≤ r,

(2.20)  cl_{Φ^{?}}(π_{01}^{∗}(α ∧ η_{A}^{j}ω_{A}^{r−j}) ∧ π_{2}^{∗}(ω_{A})) = 0 = ⟨α ∧ η_{A}^{j}ω_{A}^{r−j}, ω_{θ_{ψ}} ∧ η_{A}^{r}⟩_{X_{r}}.

In light of the definition of the map Φ_{dR}^{?∗}, equations (2.19) and (2.20) imply that

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

The proposition follows. □

Remark 2.12. We note that given a rational prime ℓ and a prime λ of F above ℓ such that F_{λ} = ℚ_{ℓ}, the maps induced by the putative correspondences Π^{?} and Φ^{?} in ℓ-adic and de Rham cohomology (the latter over F_{λ}) can be defined regardless of the existence of these correspondences, at least up to a global constant independent of λ. Indeed, let Π_{ℓ}^{∗} be any isomorphism

Π_{ℓ}^{∗} : (M_{1})_{ℓ} ≃ (M_{2})_{ℓ}

of (E_{θ_{ψ}} ⊗ℚ_{ℓ})[G_{F}]-modules. By the comparison theorem this gives rise to an E_{θ_{ψ}} ⊗_{ℚ} F_{λ}-linear isomorphism of de Rham realizations:

Π_{dR,λ}^{∗} : M_{1,dR} ⊗_{F} F_{λ}⟶M_{2,dR} ⊗_{F} F_{λ},

mapping ω_{A}^{r+1} to a (unit) E_{ψ} ⊗_{ℚ} F_{λ}-rational multiple of ω_{θ_{ψ}}. Since F_{λ} = ℚ_{ℓ}, we can rescale Π_{ℓ}^{∗} uniquely such that Π_{dR,λ}^{∗} satisfies:

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

Now as in the proof of Lemma 2.9, the Tate cycle corresponding to the normalised isomorphism Π_{ℓ}^{∗} can be viewed as a non-zero element of

(H_{et}^{1}(Ā, E_{θ_{ψ}} ⊗ ℚ_{ℓ})^{∨} ⊗ H^{2r+1}(X̄_{r}, E_{θ_{ψ}} ⊗ ℚ_{ℓ})(r))^{G_{F}} ,

and hence gives rise to a map

(2.21)  Φ_{et,λ}^{∗} : H_{et}^{2r+1}(X̄_{r}, ℚ_{ℓ})(r+1) ⊗_{ℚ} E_{θ_{ψ}}⟶H_{et}^{1}(Ā, ℚ_{ℓ})(1) ⊗_{ℚ} E_{θ_{ψ}} = V_{ℓ}(A) ⊗_{ℚ} E_{θ_{ψ}}.

By the comparison isomorphism one gets a map

(2.22)  Φ_{dR,λ}^{∗} : H_{dR}^{1}(A/F_{λ}) ⊗ E_{θ_{ψ}}⟶H_{dR}^{2r+1}(X_{r}/F_{λ}) ⊗ E_{θ_{ψ}}.

The same proof as in Proposition 2.11 shows that

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

Note that if Φ^{?} exists, then the map Φ_{dR,λ}^{∗} differs from Φ_{dR}^{?} exactly by the global constant c_{ψ,F}.

Remark 2.13. Consider the following special case (see also Section 2.7 of [BDP-cm]) in which the following assumptions are made:

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

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

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

(2.23)  ψ_{0}((a)) = ε_{K}(a mod 𝔡_{K})a,

where 𝔡_{K} = (\sqrt{−D}). The character ψ_{0} determines (uniquely, up to an isogeny) an elliptic curve A/ℚ satisfying

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

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

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

so that ψ_{A} is self-dual. Suppose that ψ = ψ_{A}^{r+1}. In this case, the setup above simplifies drastically since E_{θ_{ψ}} = ℚ and we may choose F = K. The modular parametrisation Φ^{?} arises from a class in CH^{r+1}(X_{r} ×A)(K)⊗ℚ and induces a natural transformation of functors on K-algebras:

(2.24)  Φ^{?} : CH^{r+1}(X_{r})_{0} ⊗ ℚ⟶A ⊗ ℚ.

2.5. Generalised Heegner cycles and Chow-Heegner points. Recall the notation Γ := Γ_{ε}(N) ⊂ Γ_{0}(N) in (2.4) The associated modular curve C = X_{ε}(N) has a model over ℚ obtained by realising C as the solution to a moduli problem, which we now describe. Given an abelian group G of exponent N, denote by G^{∗} the set of elements of G of order N. This set of “primitive elements” is equipped with a natural free action by (ℤ/Nℤ)^{×}, which is transitive when G is cyclic.

Definition 2.14. A Γ-level structure on an elliptic curve E is a pair (C_{N}, t), where (1) C_{N} is a cyclic subgroup scheme of E of order N,

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

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

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

Since the datum of t determines the associated cyclic group C_{N}, we sometimes drop the latter from the notation, and write (E, t) instead of (E, C_{N}, t) when convenient.

We assume now that O_{K} contains a cyclic ideal 𝔑 of norm N. Since N = D · N_{K/ℚ}(𝔣_{ψ}), this condition is equivalent to requiring that 𝔣_{ψ} is a (possibly empty) product

𝔣_{ψ} = ∏_{i} 𝔮_{i}^{n_{i}}

where 𝔮_{i} is a prime ideal in O_{K} lying over a rational prime q_{i} split in K and the q_{i} are pairwise coprime. The group scheme A[𝔑] of 𝔑-torsion points in A is a cyclic subgroup scheme of A of order N. A Γ-level structure on A of the form (A[𝔑], t) is said to be of Heegner type (associated to the ideal 𝔑).

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

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

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

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

where ϵ_{X} is the idempotent given in equation (2.2.1) of [BDP-gz], viewed as an element of the ring Corr^{0}(X_{r}, X_{r}) of algebraic correspondences from X_{r} to itself.

Definition 2.15. The Chow-Heegner point attached to the data (ψ, φ) is the point

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

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

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

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

A_{0}[𝔞] ∩ φ_{0}(A[𝔑]) = 0,

and let

φ_{𝔞} : A_{0}⟶A_{𝔞} := A_{0}/A_{0}[𝔞]

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

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

belongs to A(F) ⊗ E_{θ_{ψ}} as well.

Lemma 2.16. For all elements λ ∈ O_{c} which are prime to 𝔑, we have

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

More generally, for any 𝔟,

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

where σ_{𝔞} is the Frobenius element in Gal(F/K) attached to 𝔞.

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

Δ_{λ𝔞} − ε(λ)λ^{r}Δ_{𝔞}

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

Now pick a rational integer c prime to N and recall that we have defined in Section 3.2 of [BDP-cm] a set of Hecke characters of K, denoted Σ_{cc}(c, 𝔑, ε). (In loc. cit., we required c to be prime to pN, where p is a fixed prime split in K; however this is not a key part of the definition, and in this paper we shall pick such a p later.) The set Σ_{cc}(c, 𝔑, ε) can be expressed as a disjoint union

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

where Σ_{cc}^{(1)}(c, 𝔑, ε) and Σ_{cc}^{(2)}(c, 𝔑, ε) denote the subsets consisting of characters of infinity type (k + j, −j) with 1 − k ≤ j ≤ −1 and j ≥ 0 respectively. If p is a rational prime split in K and prime to cN, we shall denote by Σ̂_{cc}(c, 𝔑, ε) the completion of Σ_{cc}(c, 𝔑, ε) relative to the p-adic compact open topology on Σ_{cc}(c, 𝔑, ε) which is defined in Section 5.2 of [BDP-gz]. We note that the set Σ_{cc}^{(2)}(c, 𝔑, ε) of classical central critical characters “of type 2” is dense in Σ̂_{cc}(c, 𝔑, ε).

Let χ be a Hecke character of K of infinity type (r, 0) such that χN_{K} belongs to Σ_{cc}^{(1)}(c, 𝔑, ε) (so that χ is self-dual as well) and let E_{ψ,χ} denote the field generated over K by the values of ψ and χ. By Lemma 2.16, the expression

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

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

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

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

2.6. A special case. We now specialise the Chow-Heegner point construction to a simple but illustrative case, in which the hypotheses of Remark 2.13 are imposed. Thus ψ = ψ_{A}^{r+1} and the modular parametrisation Φ^{?} gives a homomorphism from CH^{r+1}(X_{r})(K) to A(K) ⊗ ℚ, We further assume

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

(2) The character χ as above is a Hecke character of infinity type (r, 0), and

χN_{K} belongs to Σ_{cc}^{(1)}(c, 𝔡_{K}, ε_{K}),

with c prime to D. The proof of Lemma 3.32 of [BDP-cm] shows that any such χ can be written as

χ = ψ_{A}^{r}χ_{0}^{−1},

where χ_{0} is a ring class character of K of conductor dividing c. Under these conditions, we have

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

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

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

where the χ_{0}-component (A(H_{c}) ⊗_{O_{K}} E_{χ})^{χ_{0}} of the Mordell-Weil group over the ring class field H_{c} is defined by

(2.27)  (A(H_{c}) ⊗_{O_{K}} E_{χ})^{χ_{0}} := {P ∈ A(H_{c}) ⊗_{O_{K}} E_{χ} such that σP = χ_{0}(σ)P, ∀σ ∈ Gal(H_{c}/K)}.

3. Chow-Heegner points over ℂ_{p}

3.1. The p-adic Abel-Jacobi map. The construction of the point P_{ψ}^{?}(χ) is only conjectural since it depends on the existence of the cycle Π^{?} and the corresponding map Φ^{?}. In order to obtain unconditional results, we will replace the conjectural map Φ^{?} by its analogue in p-adic étale cohomology.

We will let F_{0} denote the finite Galois extension of K which was denoted by F in Section 2.2. Recall that ψ ∘ N_{F_{0}/K} = ψ_{A}^{r+1}, where ψ_{A} is the Hecke character associated to an elliptic curve A/F_{0} with CM by O_{K}. Fix a rational prime p which does not divide the level N of θ_{ψ}, and such that there exists a prime v_{0} of F_{0} above p with F_{0,v_{0}} = ℚ_{p}. Recall that the choice of the place v_{0} above p in F_{0} allows us to define a normalized map

Φ_{et,p}^{∗} : H_{et}^{2r+1}(X̄_{r}, ℚ_{p}(r + 1)) ⊗_{ℚ} E_{θ_{ψ}} → H_{et}^{1}(Ā, ℚ_{p}(1)) ⊗_{ℚ} E_{θ_{ψ}} = V_{p}(A) ⊗_{ℚ} E_{θ_{ψ}} = V_{p}(A) ⊗_{K} E_{ψ}

of E_{θ_{ψ}} ⊗ ℚ_{p}[G_{F_{0}}]-modules as in equation (2.21).

Let F be any finite extension of K containing F_{0} such that the generalized Heegner cycle Δ_{φ} is defined over F. The global cohomology class

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

belongs to the pro-p Selmer group of A over F, tensored with E_{θ_{ψ}} (see [Ne2], Theorem 3.1.1.), and is defined independently of any conjectures. Furthermore, if the correspondence Φ^{?} exists, then Prop. 2.11 implies that

(3.1)  κ_{ψ}(φ) = c_{ψ,F_{0}} · δ(P_{ψ}^{?}(φ)),

where

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

is the connecting homomorphism of Kummer theory, and c_{ψ,F_{0}} is an element in (F_{0}⊗E_{θ_{ψ}})^{×} ↪ (ℚ_{p}⊗E_{θ_{ψ}})^{×}.

Let v be a place of F above v_{0}. Since κ_{ψ}(φ) belongs to the Selmer group of A over F, there is a local point in A(F_{v}) ⊗ E_{θ_{ψ}}, denoted P_{ψ}^{(v)}(φ), such that

κ_{ψ}(φ)_{|G_{Fv}} = δ_{v}(P_{ψ}^{(v)}(φ)).

More generally, as in (1.15) of the Introduction, there exists a map

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

such that

Φ_{F}^{(v)}(Δ_{φ}) = P_{ψ}^{(v)}(φ).

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

In light of Proposition 2.10 and of the construction of Chow-Heegner points given in Definition 2.15, the following conjecture is a concrete consequence of the Tate (or Hodge) conjecture for the variety X_{r} × A.

Conjecture 3.1. The local points P_{ψ}^{(v)}(φ) ∈ A(F_{v})⊗E_{θ_{ψ}} lie in Λ·(A(F)⊗E_{θ_{ψ}}), where Λ := (F_{0}⊗E_{θ_{ψ}})^{×} ↪ (ℚ_{p} ⊗ E_{θ_{ψ}})^{×}.

The goal of this chapter is to prove Conjecture 3.1 in many cases. The proof exploits the connection between the local points P_{ψ}^{(v)}(φ) and the special values of two different types of p-adic L-functions: the Katz p-adic L-function attached to K and the p-adic Rankin L-function attached to θ_{ψ} described in §3.1 and §3.2 of [BDP-cm] respectively. The reader should consult these sections for the notations and basic interpolation properties defining these two types of p-adic L-functions.

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

(3.2)  AJ_{A} : CH^{1}(A)_{0,ℚ}(F_{v})⟶Ω^{1}(A/F_{v})^{∨},

where the superscript of ∨ on the right denotes the F_{v}-linear dual. Under the identification of CH^{1}(A)_{0,ℚ}(F_{v}) with A(F_{v}) ⊗ ℚ, it is determined by the relation

(3.3)  AJ_{A}(P)(ω) = log_{ω}(P),

where ω ∈ Ω^{1}(A/F_{v}) and

log_{ω} : A(F_{v}) ⊗ ℚ⟶F_{v}

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

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

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

attached to the variety X_{r}, where Fil^{j} refers to the j-th step in the Hodge filtration on algebraic de Rham cohomology. Details on the definition of AJ_{X_{r}} can be found in Section 3 of [BDP-gz], where it is explained how AJ_{X_{r}} can be calculated via p-adic integration.

In light of Remark 2.12, the functoriality of the Abel-Jacobi maps is expressed in the following commutative diagram relating AJ_{A} and AJ_{X_{r}}:

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

Proposition 3.2. For all isogenies φ : (A, t_{A}, ω_{A})⟶(A′, t′, ω′) of elliptic curves with Γ-level structure,

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

Proof. By equation (3.3) and the definition of P_{ψ}^{(v)}(φ),

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

The commutative diagram (3.4) shows that

(3.6)  AJ_{A}(Φ_{F}^{(v)}(Δ_{φ}))(ω_{A}) = AJ_{X_{r}}(Δ_{φ})(Φ_{dR,v}^{∗}(ω_{A})) = AJ_{X_{r}}(Δ_{φ})(ω_{θ_{ψ}} ∧ η_{A}^{r}).

Proposition 3.2 now follows from (3.5) and (3.6). □

We will study the local points P_{ψ}^{(v)}(φ) via the formula of Proposition 3.2.

3.2. Rationality of Chow-Heegner points over ℂ_{p}. We begin by placing ourselves in the setting of Section 2.6, in which

ψ = ψ_{A}^{r+1}, χ = ψ_{A}^{r}χ_{0}

where χ_{0} is a ring class character of K of conductor c. In this case, we can take F_{0} = K. Let p be a prime split in K and fix a prime 𝔭 of K above p. We set

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

the latter being defined analogously to (2.26). The next theorem is one of the main results of this paper.

Theorem 3.3. There exists a global point P_{A,r}(χ_{0}) ∈ (A(H_{c}) ⊗_{O_{K}} E_{χ})^{χ_{0}} satisfying

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

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

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

Proof. By Proposition 3.2,

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

for an explicit cycle Δ_{ψ}(χ) ∈ CH^{r+1}(X_{r})_{0} ⊗ E_{χ}. Theorem 5.13 of [BDP-gz] with f = θ_{ψ} and j = 0 gives

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

where L_{p}(θ_{ψ}, χN_{K}) and Ω_{p}(A) are respectively the p-adic Rankin L-function attached to θ_{ψ} and the padic period attached to A as described in Sections 3.2 and 2.4 of [BDP-cm]. The fact that θ_{ψ} has Fourier coefficients in ℚ and that its Nebentype character ε_{K} is trivial when restricted to K implies that the field E_{ψ,χ,ε_{K}} occurring in Corollary 3.17 of Section 3.4 of [BDP-cm] is equal to E_{χ}. Therefore, this corollary implies that

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

where the factors ℒ_{p,c𝔡_{K}} (ψ^{−1}χN_{K}) and ℒ_{p,c𝔡_{K}} (ψ^{∗−1}χN_{K}) are values of the Katz two-variable p-adic L-function with conductor c𝔡_{K}, following the notations that are adopted in Section 3.1 of [BDP-cm]. The character ν^{∗} = ψ_{A}^{∗}χ_{0} lies in the region Σ_{sd}^{(1)}(c𝔡_{K}) described in Section 3.1 of loc. cit. and is of type (0, 1). Hence, Theorem 3.30 of Section 3.6 of loc. cit. can be invoked. This theorem gives a global point P_{A}(χ_{0}) ∈ (A(H_{c}) ⊗_{O_{K}} E_{χ})^{χ_{0}} which is of infinite order if and only if L′(ψ_{A}χ_{0}^{−1}, 1) ≠ 0, and satisfies

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

Furthermore, the character ψ_{A}^{2r+1}χ_{0}N_{K}^{−r} belongs to the domain Σ_{sd}^{(2)}(c𝔡_{K}) of classical interpolation for the Katz p-adic L-function. Proposition 2.15 and Lemma 2.14 in Section 2.3 of [BDP-cm] show that the p-adic period attached to this central critical character is given by

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

Corollary 3.3 of Section 3.1 of loc. cit. then implies that, up to multiplication by a non-zero element of E_{χ},

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

After setting

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

equations (3.10) and (3.12) can be used to rewrite (3.9) as

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

Theorem 3.3 now follows by combining (3.7), (3.8) and (3.14). □

We now state a more general, but less precise, version of Theorem 3.3. Let ψ and χ be two self-dual characters of K of infinity types (r + 1, 0) and (r, 0) respectively, as in Sec. 2.5. Let F_{ψ,χ} be the subfield of ℚ̄ generated over K by F and E_{ψ,χ}, and let ν := ψχ^{−1}, so that ν is a self-dual Hecke character of K of infinity type (1, 0) attached to the pair (ψ, χ).

Theorem 3.4. There exists a global point P_{ψ}(χ) ∈ A(F) ⊗_{O_{K}} E_{ψ,χ} such that

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

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

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

Proof. The proof proceeds along the same lines as (but is simpler than) the proof of Theorem 3.3. This earlier proof applies to a more special setting but derives a more precise result, in which it becomes necessary to keep a more careful track of the fields of scalars involved. To prove Theorem 3.4, it suffices to rewrite the proof of Theorem 3.3 with E_{χ}^{×} replaced by F_{ψ,χ}^{×} and (ψ_{A}^{r+1}, ψ_{A}^{r}χ_{0}) replaced by (ψ, χ). Note that equations (3.10) and (3.11) hold modulo the larger group F_{ψ,χ}^{×} without the Gauss sum factors which can therefore be ignored. □

We now specialise the setting of Theorem 3.3 even further by assuming that χ_{0} = 1 is the trivial character, so that ψ = ψ_{A}^{r+1} and χ = ψ_{A}^{r}, and set

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

In this case, the coefficient field E_{χ} is equal to K, and Theorem 3.3 asserts the existence of a point P_{A,r} ∈ A(K) ⊗ ℚ such that

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

It is instructive to refine the argument used in the proof of Theorem 3.3 to resolve the ambiguity by the non-zero scalar in K^{×}, in order to examine the dependence on r of the local point P_{A,r}^{(𝔭)}. This is the content of the next result.

Theorem 3.5. For all odd r ≥ 1, the Chow-Heegner point P_{A,r}^{(𝔭)} belongs to A(K) ⊗ ℚ and is given by the formula

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

where ℓ(r) ∈ ℤ satisfies

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

and P_{A} is a generator of A(K) ⊗ ℚ depending only on A but not on r.

Proof. As in the proof of Theorem 3.3, we combine (3.7) and Theorem 5.13 of [BDP-gz] with (f, j) = (θ_{ψ_{A}^{r+1}}, 0) and χN_{K} = ψ_{A}^{r}N_{K} playing the role of χ, to obtain

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

Since χ(𝔭̄) = ψ_{A}(𝔭̄)^{r} and a_{p}(θ_{ψ}) = ψ_{A}^{r+1}(𝔭̄) + ψ_{A}^{r+1}(𝔭), the Euler factor appearing in (3.16) is given by

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

Therefore,

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

On the other hand, by Corollary 2.17 of [BDP-cm] with c = 1 and j = 0

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

where we write ℒ_{p} for ℒ_{p,𝔡_{K}}. By Lemma 5.3 of [BDP-gz], the norm 1 scalar w(θ_{ψ}, χ) belongs to K, and is only divisible by the primes above \sqrt{−D}. Therefore it is a unit in O_{K}, and hence is equal to ±1. We obtain

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

Let P_{A} = P_{A}(1) ∈ A(K) ⊗ ℚ be as in (3.10), but chosen specifically so that

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

By the interpolation property for the Katz L-function given, for instance, in Proposition 3.5 of Section 3.1 of [BDP-cm] with j = r and ν = ψ_{A}^{2r+1}N_{K}^{−r} = ψ_{A}^{r+1}ψ_{A}^{∗−r},

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

After substituting equations (3.20) and (3.21) into (3.19), and using the fact that

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

we find

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

Hence, by (3.17), we obtain

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

The result follows since ℓ(r) is shown to be an integer in [RV]. □

4. Chow-Heegner points over ℂ

4.1. The complex Abel-Jacobi map. For simplicity, we will confine ourselves in this section to working under the hypotheses that were made in Remark 2.13 where K is assumed in particular to have discriminant −D, with

D ∈ S := {7, 11, 19, 43, 67, 163}.

Let us suppose for the moment that an algebraic correspondence Π^{?} ∈ CH^{r+1}(W_{r} × A^{r+1}) ⊗ ℚ as in Proposition 2.10 exists. By taking an integer multiple of this correspondence we may assume that it has integer coefficients. As before then, viewing it as a correspondence

Φ^{?} ∈ CH^{r+1}(X_{r} × A),

where X_{r} = W_{r} × A^{r}, we get a modular parametrization also denoted Φ^{?}:

Φ^{?} : CH^{r+1}(X_{r})_{0}⟶ CH^{1}(A)_{0} = A.

By Propositions 2.10 and 2.11, we have (with ψ := ψ_{A}^{r+1})

(4.1)  Π_{dR}^{?∗}(ω_{A}^{r+1}) = c_{ψ,K} · ω_{θ_{ψ}},  Φ_{dR}^{?∗}(ω_{A}) = c_{ψ,K} · ω_{θ_{ψ}} ∧ η_{A}^{r},

for some scalar c_{ψ,K} ∈ K^{×}. This scalar can be viewed as playing the role of the Manin-constant in the context of the modular parametrisation of A by CH^{r+1}(X_{r})_{0}.

Question 4.1. When is it possible to choose an integral cycle Π^{?} so that c_{ψ,K} = 1?

The difficulty in computing the modular parametrisation Φ^{?} and the resulting Chow-Heegner points arises from the fact that it is hard in general to explicitly produce the correspondence Φ^{?}, or even to prove its existence. In this section we shall see that it is possible to define a complex avatar Φ_{ℂ} of Φ^{?} unconditionally and compute it numerically to great precision in several examples. Note that if the cycle Φ^{?} exists, then equation (4.1) shows that c_{ψ,K} · ω_{θ_{ψ}} ∧ η_{A}^{r+1} is an integral Hodge class on W_{r} × A^{r+1}. The construction of Φ_{ℂ} is based on the observation that one can show the following independently using a period computation, as in [Scha] Ch. 5, Thm 2.4.

Proposition 4.2. There exists a scalar c_{r} ∈ K^{×} such that Ξ := c_{r} · ω_{θ_{ψ}} ∧ η_{A}^{r+1} is an integral Hodge class on W_{r} × A^{r+1}.

Let us fix such a scalar c_{r} ∈ K^{×}. Clearly we may assume that c_{r} is in fact in O_{K}. Let

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

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

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

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

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

defined just as in (4.3), but where AJ_{X_{r}}^{∞}(Δ)(ω) is now defined by integrating any smooth representative of the de Rham cohomology class ω against a (2r + 1)-chain on X_{r}(ℂ) having Δ as boundary. (Cf. the description in Section 4 of [BDP-caj] for example.) The map AJ_{X_{r}}^{∞} is the complex analogue of the p-adic Abel-Jacobi map AJ_{X_{r}} that was introduced and studied in Section 3.

If the Hodge conjecture holds, there is an algebraic cycle Φ^{?} = Π^{?} ∈ CH^{r+1}(X_{r} × A) ⊗ ℚ whose cohomology class equals Ξ. If further Φ^{?} has integral coefficients, then we have the following commutative diagram which is the complex counterpart of (3.4) and which expresses the functoriality of the Abel-Jacobi maps under correspondences:

(4.5)  \xymatrix{CH^{r+1}(X_{r})_{0}(ℂ) \ar@{.>}[d]^{Φ_{ℂ}^{?}} \ar[r]^{^{AJ{∞ \atop Xr}}} & Fil^{r+1} H_{dR}^{2r+1}(X_{r}/ℂ)^{∨}/ Im H_{2r+1}(X_{r}(ℂ), ℤ) \ar[d]_{(Φ_{dR,ℂ}^{∗})^{∨}} \\ CH^{1}(A)_{0}(ℂ) \ar[r]^{_{AJ{∞ \atop A}}} & Ω^{1}(A/ℂ)^{∨}/ Im H_{1}(A(ℂ), ℤ),}

where the map Φ_{dR,ℂ}^{∗} is defined to be the one induced by the integral Hodge class Ξ. Note that by construction

Φ_{dR,ℂ}^{∗}(ω_{A}) = c_{r} · ω_{θ_{ψ}} ∧ η_{A}^{r}.

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

We will now discuss how the map Φ_{ℂ} can be computed in practice. Recall the distinguished element ω_{A} of Ω^{1}(A/ℂ) and let

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

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

φ(t_{A}) = t′ and φ^{∗}(ω′) = ω_{A}.

The following proposition, which is the complex counterpart of Proposition 3.2, expresses the Abel-Jacobi image of the complex point P_{ψ}(φ) := Φ_{ℂ}(Δ_{φ}) in terms of the Abel-Jacobi map on X_{r}.

Proposition 4.3. For all isogenies φ : (A, t_{A}, ω_{A})⟶(A′, t′, ω′) of elliptic curves with Γ-level structure,

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

Proof. The proof is the same as for Proposition 3.2. By definition of P_{ψ}(φ) combined with the commutative diagram (4.5),

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

Since Φ_{dR,ℂ}^{∗}(ω_{A}) = c_{r} · ω_{θ_{ψ}} ∧ η_{A}^{r}, Proposition 4.3 follows. □

Remark 4.4. In the above proposition and elsewhere below, we assume that Δ_{φ} has been multiplied by a nonzero integer so as to have integral coefficients.

We now turn to giving an explicit formula for the right hand side of the equation in Proposition 4.3. To do this, let Λ_{ω′} ⊂ ℂ be the period lattice associated to the differential ω′ on A′. Note that Λ_{ω_{A}} is contained in Λ_{ω′} with index deg(φ).

Definition 4.5. A basis (ω_{1}, ω_{2}) of Λ_{ω′} is said to be admissible relative to (A′, t′) if (1) The ratio τ := ω_{1}/ω_{2} has positive imaginary part;

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

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

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

Proposition 4.6. Let Δ_{φ} be the generalised Heegner cycle corresponding to the isogeny

φ : (A, t_{A}, ω_{A})⟶(A′, t′, ω′)

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

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

Proof. We begin by observing that replacing ω_{A} by a scalar multiple λω_{A} multiplies both the left and right hand sides of (4.6) by λ^{−r}. Hence we may assume, after possibly rescaling Λ_{ω′}, that the admissible basis (ω_{1}, ω_{2}) is of the form (2πiτ, 2πi) with τ ∈ H. The case j = 0 in Theorem 8.2 of [BDP-caj] then implies that

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

The proposition follows. □

Theorem 4.7. Let P_{ψ}(φ) be the Chow-Heegner point corresponding to the generalised Heegner cycle Δ_{φ}. With notations as in Proposition 4.6,

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

Proof. This is an immediate corollary of Propositions 4.3 and 4.6. □

In the following, we shall describe some numerical evidence for the rationality of the points P_{ψ}(φ). Since the constant c_{r} lies in O_{K} \ {0} and since A has CM by O_{K}, it will suffice in the following to show rationality assuming c_{r} = 1.

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

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

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

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

Table 1: The canonical elliptic curve A

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

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

and (ω_{1}, ω_{2}) is an admissible basis for Λ_{A} in the sense of Definition 4.5. The elliptic curve A has Mordell-Weil rank 0 over ℚ when D = 7 and rank one otherwise. A specific generator P_{A} for A(ℚ) ⊗ ℚ is given in the last column of Table 1.

4.2.1. Chow-Heegner points of level 1. For D ∈ S := {11, 19, 43, 67, 163}, the elliptic curve A has rank 1 over ℚ. Let r ≥ 1 be an odd integer. As remarked above, it suffices to check rationality assuming c_{r} = 1. By Theorem 4.7, the Chow-Heegner point P_{A,r} attached to the class of the diagonal Δ ⊂ (A × A)^{r} is given by

(4.9)  AJ_{A}^{∞}(P_{A,r})(ω_{A}) = J_{r} := ω_{2}^{−r} \frac{(2πi)^{r+1}}{(τ − τ̄)^{r}} ∫_{i∞}^{τ}(z − τ̄)^{r}θ_{ψ}(z)dz,

where (ω_{1}, ω_{2}) is the admissible basis of Λ_{A} given in (4.8), and τ = \frac{ω_{1}}{ω_{2}} = \frac{D+\sqrt{−D}}{2D}. Hence the complex point P_{A,r} can be computed as the natural image of the complex number J_{r} under the Weierstrass uniformisation.

We have calculated the complex points P_{A,r} for all D ∈ S and all r ≤ 15, to roughly 200 digits of decimal accuracy. The calculations indicate that

(4.10)  P_{A,r} =^{?} \sqrt{−D} · m_{r} · P_{A} (mod A(ℂ)[ι_{r}]),

where P_{A} is the generator of A(ℚ)⊗ℚ given in Table 1, ι_{r} is a small integer, and m_{r} is the rational integer listed in Table 2 below, in which the columns correspond to D ∈ S and the rows to the odd r between 1 and 15.

11  19  43  67  163
1  1  1  1  1  1
3  2  6  36  114  2172
5  −8  −16  440  6920  3513800
7  14  −186  −19026  −156282  3347376774
9  304  4176  −8352  −34999056  −238857662304
11 −352 −33984  33708960  3991188960  −3941159174330400
13 76648  545064 −2074549656  46813903656  1904546981028802344
15 274736 40959504 47714214240 −90863536574160 8287437850155973464480

Table 2: The constants m_{r} for 1 ≤ r ≤ 15.

The first 6 lines in this table, corresponding to 1 ≤ r ≤ 11, are in perfect agreement with the values that appear in the third table of Section 3.1 of [RV]. This coincidence, combined with Theorem 3.1. of [RV], suggests the following conjecture which is consistent with the p-adic formulae obtained in Theorem 3.5.

Conjecture 4.8. For all D ∈ S and all odd r ≥ 1, the Chow-Heegner point P_{A,r} belongs to A(K) ⊗ ℚ and is given by the formula

(4.11)  P_{A,r} = \sqrt{−D} · m_{r} · P_{A},

where m_{r} ∈ ℤ satisfy the formula

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

and P_{A} is the generator of A(ℚ) ⊗ ℚ given in Table 1.

The optimal values of ι_{r} that were observed experimentally are recorded in Table 3 below, for 1 ≤ r ≤ 31.

\begin{matrix}r & 11 & 19 & 43 & 67 & 163 & r & 11 & 19 & 43 & 67 & 163 \\ 1 & 3 & 1 & 1 & 1 & 1 & 17 & 3^{3} & 7 & 1 & 19 & 1 \\ 3 & 3 · 5 & 5 & 1 & 1 & 1 & 19 & 3 · 5^{2} & 5^{2} · 11 & 11 & 1 & 1 \\ 5 & 2 · 3^{2} & 2 · 7 & 2 & 2 & 2 & 21 & 3 · 23 & 23 & 23 & 23 & 1 \\ 7 & 2 · 7 & 5 & 1 & 1 & 1 & 23 & 3^{2} · 5 & 5 · 7 & 13 & 1 & 1 \\ 9 & 3 & 11 & 11 & 1 & 1 & 25 & 3 & 1 & 1 & 1 & 1 \\ 11 & 3^{2} · 5 & 5 · 7 & 13 & 1 & 1 & 27 & 3 · 5 & 5 & 1 & 29 & 1 \\ 13 & 3 & 1 & 1 & 1 & 1 & 29 & 3^{2} · 31 & 7 · 11 & 11 · 31 & 1 & 1 \\ 15 & 3 · 5 & 5 · 17 & 17 & 17 & 1 & 31 & 3 · 5 & 5 · 17 & 17 & 1 & 1 \end{matrix}

Table 3: The ambiguity factor ι_{r} for 1 ≤ r ≤ 31.

Remark 4.9. The data in Table 3 suggests that the term ι_{r} in (4.10) is only divisible by primes that are less than or equal to r + 2. One might therefore venture to guess that the primes ℓ dividing ι_{r} are only those for which the mod ℓ Galois representation attached to ψ_{A}^{r+1} has very small image, or perhaps non-trivial G_{K}-invariants.

4.2.2. Chow-Heegner points of prime level. We may also consider (for a fixed D and a fixed odd integer r) the Chow-Heegner points on A attached to non-trivial isogenies φ. For instance, let ℓ ≠ D be a prime. There are ℓ + 1 distinct isogenies φ_{j} : A⟶A′_{j} of degree ℓ (with j = 0, 1, …, ℓ − 1, ∞) attached to the lattices Λ′_{0}, …, Λ′_{ℓ−1}, Λ′_{∞} containing Λ_{A} with index ℓ. These lattices are generated by the admissible bases

Λ′_{j} = ℤ (\frac{ω_{1} + jω_{2}}{ℓ} ) ⊕ ℤω_{2}, Λ′_{∞} = ℤω_{1} ⊕ ℤ\frac{ω_{2}}{ℓ} .

The elliptic curves A′_{j} and the isogenies φ are defined over the ring class field H_{ℓ} of K of conductor ℓ. Let

J_{r}(ℓ, j) := ℓ^{r}ω_{2}^{−r} \frac{(2πi)^{r+1}}{(τ − τ̄)^{r}} ∫i∞^{\frac{τ+j}{ℓ}} (z − \frac{τ̄ + j}{ℓ} )^{r} θ_{D,r}(z)dz, 0 ≤ j ≤ ℓ − 1,
J_{r}(ℓ, ∞) := ε_{K}(ℓ)ω_{2}^{−r} \frac{(2πi)^{r+1}}{(τ − τ̄)^{r}} ∫_{i∞}^{ℓτ} (z − ℓτ̄)^{r}θ_{ψ}(z)dz

be the associated complex invariants and let P_{A,r}(ℓ, j) and P_{A,r}(ℓ, ∞) denote the corresponding points in ℂ/Λ_{A} = A(ℂ).

We have attempted to verify the following conjecture numerically.

Conjecture 4.10. For all ℓ ≠ D and all j ∈ P_{1}(𝔽_{ℓ}), some (nonzero) multiple of the complex points P_{A,r}(ℓ, j) belong to the Mordell-Weil group A(H_{ℓ}).

We have tested this prediction numerically for r = 1 and all

D ∈ S, ℓ = 2, 3, 5, 7, 11,

as well as in a few cases where r = 3. Such calculations sometimes required several hundred digits of numerical precision, together with a bit of trial and error. The necessity for this arose because Conjecture 4.10 only predicts that some multiple of the points P_{A,r}(ℓ, j) belong to A(H_{ℓ}), as one would expect from Remark 4.4 as well as the possibility that the constant c_{r} is not 1. One finds in practice that these complex points do need to be multiplied by a (typically small) integer in order to belong to A(H_{ℓ}). Furthermore, the resulting global points appear (as suggested by (4.11) in the case ℓ = 1) to be divisible by \sqrt{−D}, and this causes their heights to be rather large. It is therefore better in practice to divide the P_{A,r}(ℓ, j) by \sqrt{−D}, which introduces a further ambiguity of A(ℂ)[\sqrt{−D}] in the resulting global point. The conjecture that was eventually tested numerically is the following non-trivial strengthening of Conjecture 4.10:

Conjecture 4.11. Given integers n ∈ ℤ^{≥1} and 0 ≤ s ≤ D − 1, let

J′_{r}(ℓ, j) = n · \frac{J_{r}(ℓ, j) − sω_{1}}{\sqrt{−D}} ,  0 ≤ j ≤ ℓ − 1,
J′_{r}(ℓ, ∞) = n · \frac{J_{r}(ℓ, ∞) − sε_{K}(ℓ)ℓ^{r}ω_{1}}{\sqrt{−D}} ,

and let P′_{A,r}(ℓ, j) ∈ A(ℂ) be the associated complex points. Then there exist n = n_{D,r} and s = s_{D,r}, depending on D and r but not on ℓ and j, for which the points P′_{r}(ℓ, j) belong to A(H_{ℓ}) and satisfy the following:

(1) If ℓ is inert in K, then Gal(H_{ℓ}/K) acts transitively on the set

{P′_{A,r}(ℓ, j), j ∈ P_{1}(𝔽_{ℓ})}

of Chow-Heegner points of level ℓ.

(2) If ℓ = λλ̄ is split in K, then there exist j_{1}, j_{2} ∈ P_{1}(𝔽_{ℓ}) for which

P′_{A,r}(ℓ, j_{1}) = ε_{K}(λ)λ^{r}P′_{A,r}, P′_{A,r}(ℓ, j_{2}) = ε_{K}(λ̄)λ̄^{r}P′_{A,r},

and Gal(H_{ℓ}/K) acts transitively on the remaining set

{P′_{A,r}(ℓ, j), j ∈ P_{1}(𝔽_{ℓ}) − {j_{1}, j_{2}}}

of Chow-Heegner points of level ℓ.

We now describe a few sample calculations that lend support to Conjecture 4.11.

1. The case D = 7. Consistent with the fact that the elliptic curve A has rank 0 over ℚ (and hence over K as well), the point P_{A,r} appears to be a torsion point in A(ℂ), for all 1 ≤ r ≤ 31. For example, the invariant J_{1} agrees with (ω_{1} + ω_{2})/8 to the 200 decimal digits of accuracy that were calculated. When ℓ = 2, it also appears that the quantities J_{1}(2, j) belong to \frac{1}{8}Λ_{7}. There is no reason, however, to expect the Chow-Heegner points P_{A,r}(ℓ, j) to be torsion for larger values of ℓ. Experiments suggest that the constants in Conjecture 4.11 are

n_{7,1} = 4, s_{7,1} = 0.

For example, when ℓ = 3, the ring class field of conductor ℓ is a cyclic quartic extension of K containing K(\sqrt{21}) as its quadratic subfield. In that case, the points P′_{A,1}(3, j) satisfy

P′_{A,1}(3, 0) = P′_{A,1}(3, 1) = −P′_{A,1}(3, 2) = −P′_{A,1}(3, ∞),

and agree to 600 digits of accuracy with a global point in A(ℚ(\sqrt{21})) of relatively small height, with x-coordinate given by

x = \frac{259475911175100926920835360582209388259}{41395589491845015952295204909998656004}.

2. The case D = 19. To compute the Chow-Heegner points of conductor 3 in the case D = 19 and r = 1, it appears that one can take

n_{19,1} = 1, s_{19,1} = 1.

Perhaps because of the small value of n_{19,1}, the points P′_{A,1}(ℓ, j) appear to be of relatively small height and can easily be recognized as global points, even for moderately large values of ℓ. For instance, the points P′_{A,1}(3, j) seem to have x-coordinates of the form

x = \frac{−19 ± 3\sqrt{57}}{2},

and their y-coordinates satisfying the degree 4 polynomial

x^{4} + 2x^{3} + 8124x^{2} + 8123x − 217886

whose splitting field is the ring class field H_{3} of K of conductor 3.

When ℓ = 7, which is split in K/ℚ, the ring class field H_{7} is a cyclic extension of K of degree 6. It appears that the points P′_{A,1}(7, 3) and P′_{A,1}(7, 5) belong to A(K) and are given by

P′_{A,1}(7, 3) = \frac{3 + \sqrt{−19}}{2}P_{A}, P′_{A,1}(7, 5) = \frac{3 − \sqrt{−19}}{2}P_{A}.

The 6 remaining points are grouped into three pairs of equal points,

P′_{A,1}(7, 0) = P′_{A,1}(7, 2),  P′_{A,1}(7, 1) = P′_{A,1}(7, 6),  P′_{A,1}(7, 4) = P′_{A,1}(7, ∞),

whose x and y coordinates appear to satisfy the cubic polynomials

9x^{3} + 95x^{2} + 19x − 1444, 27x^{3} − 235x^{2} + 557x + 1198

respectively. The splitting field of both of these polynomials turns out to be the cubic subfield L of the ring class field of K of conductor 7. One obtains as a by-product of this calculation 3 independent points in A(L) which are linearly independent over O_{K}. We expect that these three points give a K-basis for A(L) ⊗ ℚ (and therefore that A(L) has rank 6) but have not checked this numerically.

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