Cycles on modular varieties and rational points on elliptic curves

Henri Darmon

July 31, 2009

This is a summary of a three-part lecture series given at the meeting on “Explicit methods in number theory” that was held in Oberwolfach from July 12 to 18, 2009. The theme of this lecture series was the explicit construction of algebraic points on elliptic curves from cycles on modular varieties. Given a fixed elliptic curve E over ℚ, the goal is to better understand the group E(ℚ̄) of algebraic points on E by focusing on the following question:

Which points in E(ℚ̄) can be accounted for by a “modular construction”?

Heegner points arising from CM points on modular curves are the prototypical example of such a modular construction. While we do not dispose of a completely satisfactory general definition of modular points, fulfilling the conflicting requirements of flexibility and mathematical precision, several “test cases” that go beyond the setting of Heegner points have been studied over the last 10 years (cf. [Da01], [DL], [BDG], [Da04], [Tr], [Gre], [BDP2]). Three illustrative examples were touched upon in these lectures:

1. [BDP1], [BDP2]. “Chow-Heegner points” arising from algebraic cycles on higher dimensional varieties. The existence and key properties of Chow-Heegner points are typically conditional on the Hodge or Tate conjectures on algebraic cycles.

2. [DL], [BDG], [CD]. “Stark-Heegner points” arising from ATR (“Almost Totally Real”) cycles on Hilbert modular varieties parametrising elliptic curves over totally real fields. These ATR cycles are not algebraic, and the expected algebraicity properties of the associated Stark-Heegner points do not seem (for now) to be part of a systematic philosophy.

3. [Da01], [DP]. Stark-Heegner points attached to real quadratic cycles on the “mock Hilbert modular surface” SL_{2}(ℤ[1/p])\(H_{p} ×H) parametrising an elliptic curve E over ℚ of prime conductor p. These real quadratic cycles are indexed by ideal classes of orders in a real quadratic field K, and are topologically isomorphic to ℝ/ℤ. By an analytic process that combines complex and p-adic integration, they can be made to yield p-adic points on E which are expected to be defined over class fields of K. This setting leads to convincing experimental evidence for the existence of a theory of “complex multiplication for real quadratic fields”.

1 Heegner Points

We begin with a brief sketch of the classical picture which we aim to generalize.

Modular parametrisations. Let E be an elliptic curve over ℚ, and let N be its conductor. The classical construction of Heegner points is based on the modularity theorem of [Wi], [TW], as completed in [BCDT]. It asserts that

L(E, s) = L(f, s),  (1)

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 E is established by showing that the p-adic Galois representation

V_{p}(E) = (lim_{←,n} E[p^{n}]) ⊗ ℚ_{p} = H_{et}^{1}(Ē, ℚ_{p})(1)  (2)

is a constituent of the first p-adic étale cohomology of the modular curve X_{0}(N). The surjective G_{ℚ}-equivariant projection of Galois representations

H_{et}^{1}(\overline{X_{0}(N)}, ℚ_{p})⟶H_{et}^{1}(Ē, ℚ_{p})  (3)

gives rise to a non-trivial Tate cycle

Π_{p} ∈ H_{et}^{2}(\overline{X_{0}(N) × E}, ℚ_{p})(1)^{G_{ℚ}}.  (4)

By the Tate conjecture for curves over number fields that was proved by Faltings, there is therefore a non-constant morphism over ℚ

Φ : J_{0}(N)⟶E,  (5)

where J_{0}(N) is the Jacobian of X_{0}(N). This stronger, “geometric” form of modularity is crucial for the Heegner point construction.

CM points. The modular curve X_{0}(N) is equipped with a distinguished supply of 0dimensional cycles CM_{K} ⊂ Div^{0}(X_{0}(N)(K^{ab})) attached to any imaginary quadratic field K. The group CM_{K} consists of degree zero divisors supported on CM points attached to the moduli of elliptic curves with complex multiplication by an order in K. It is not hard to show that Φ(CM_{K}) is an infinitely generated subgroup of E(K^{ab}); it will be referred to as the group of Heegner points on E attached to K. The importance of Heegner points can be justified on (at least) three grounds.

1. The Gross-Zagier formula [GZ] relates the heights of certain points in Φ(CM_{K}) to the central critical derivatives of the Hasse-Weil L-series of E over K, twisted by abelian characters of K, and thus supplies a link between the arithmetic of E and its Hasse-Weil L-series.

2. Following a method of Kolyvagin (cf. [Gr2]), the non-triviality of certain Heegner points can be used to bound the Selmer group of E (and therefore, its rank and Shafarevich-Tate group). Combined with the Gross-Zagier formula, this has led to the strongest known results on the Birch and Swinnerton-Dyer conjecture, most notably the theorem that

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

3. Heegner points can be computed efficiently in practice by analytic methods. After identifying the set Y_{0}(N)(ℂ) of complex points on the open modular curve with the quotient Γ_{0}(N)\H, and replacing E(ℂ) by the isogenous torus ℂ/Λ_{f} for an appropriate period lattice Λ_{f} attached to f, one has

Φ(τ) = ∫_{i∞}^{τ} 2πif(z)dz = ∑_{n=1}^{∞} \frac{a_{n}}{n} e^{2πinτ} (mod Λ_{f}).

This formula leads to efficient algorithms for computing Heegner points numerically, which have been implemented in software systems like Pari-GP, Magma, and SAGE.

2 Chow-Heegner points

Chow Groups. Given a variety V of dimension d defined over a field F, let

CH^{j}(V)(F) = \{ \begin{matrix}Codimension j algebraic cycles on V over F \\ modulo rational equivalence \end{matrix} \} ,
CH^{j}(V)_{0}(F) = the subgroup of null-homologous cycles.

Modular parametrisations. Any element Π of the Chow group CH^{d+1−j}(V × E)(ℚ) induces homomorphisms

Φ_{F} : CH^{j}(V)_{0}(F)⟶E(F)  (6)

for any F ⊃ ℚ, by the rule

Φ_{F} (Δ) := π_{E}(π_{V}^{−1}(Δ̃) · Π̃),  (7)

where π_{V} and π_{E} denote the natural projections from V × E to V and E respectively and Δ̃ and Π̃ are representatives of the class of Δ and Π, chosen so that π_{V}^{−1}(Δ̃) and Π̃ intersect transversally. The assignment Φ : F ↦ Φ_{F} is a natural transformation from CH^{j}(V)_{0} to E, viewed as functors on ℚ-algebras. This leads to the following informal definition:

Definition 2.1. A modular parametrisation of E is a triple (V, Π, j) where

1. V is a “modular variety” of dimension d;

2. Π is a cycle class in CH^{d+1−j}(V × E)(ℚ);

3. the induced morphism Φ : CH^{j}(V)_{0}⟶E is non trivial.

The non-triviality condition on Φ merits some clarification. The most obvious notion of nontriviality is to require the existence of a cycle Δ ∈ CH^{j}(V)_{0}(ℚ̄) for which Φ(Δ) is non-zero in E(ℚ̄) ⊗ ℚ. A second notion rests on the fact that the correspondence Π induces a functorial map on deRham cohomology:

Φ_{dR}^{∗} : H_{dR}^{1}(E/ℚ)⟶H_{dR}^{2d−2j+1}(V/ℚ).

The modular parametrisation Φ will be said to be non-trivial if the class of Φ_{dR}^{∗}(ω_{E}) is nonzero, where ω_{E} is a non-zero regular differential on E. We will henceforth work with this cohomological notion of non-triviality.

Modular varieties. Definition 2.1 above falls short of being mathematically precise because we have not explained what is meant by “modular variety”. Loosely speaking, such a variety is one which can be related to a Shimura variety in a reasonably direct way. For instance, a Shimura variety is a modular variety, as is the universal object or the r-fold fiber product of the universal object over a Shimura variety of PEL type. Examples include modular and Shimura curves, Kuga-Sato varieties, Hilbert modular varieties, Siegel modular varieties, Shimura varieties attached to the the orthogonal group O(2, n) or the unitary group U(p, q), etc. For the purposes of these lectures, the term “modular variety” is best interpreted informally in the broadest possible sense, as any variety whose cohomology is related to modular forms.

Chow-Heegner points. Modular varieties frequently contain a plentiful supply of arithmetically interesting algebraic cycles. The images in E(ℚ̄) of such special cycles under a modular parametrisation can be viewed as “higher-dimensional” analogues of Heegner points: they will be referred to as Chow-Heegner points.

The general program. Given an elliptic curve E, it would be of interest to construct modular parametrisations to E in the greatest possible generality, study their basic properties, and explore the relations (if any) between the resulting systems of Chow-Heegner points and values of L-series attached to E.

3 Generalised Heegner cycles

We flesh out the loosely formulated program of the previous paragraph in a simple but non-trivial setting, in which E = A is an elliptic curve with complex multiplication by an imaginary quadratic field K, and V is a suitable family of 2r-dimensional abelian varieties fibered over a modular curve. This construction (to which two of the lectures in the series ended up being devoted to) is part of a work in progress with Massimo Bertolini and Kartik Prasanna [BDP1], [BDP2].

The setting. Fix a quadratic imaginary field K, and let A be an elliptic curve with complex multiplication by the maximal order in K. In order to simplify the presentation of the main results, we make the following assumptions:

Assumption 3.1. 1. The field K has class number one, unit group of order two, and odd discriminant. This implies that D := − Disc(K) is one of the following 6 primes:

D = 7, 11, 19, 43, 67, or 163.

2. The elliptic curve A is defined over ℚ and has conductor D^{2}.

These assumptions are of course very restrictive; they are only made to ease the exposition, and the main results of [BDP1] and [BDP2] are obtained under more general conditions in which K is not assumed to have class number one.

Let ϵ_{D} : (ℤ/Dℤ)^{×}⟶ ± 1 be the quadratic Dirichlet character of conductor D attached to K. Because A has complex multiplication, its modularity follows from the fact (known much before the work of Wiles, of course) that

L(A, s) = L(ψ, s),

where ψ is the Hecke character of K of infinity type (1, 0) defined on (principal) ideals by the rule

ψ((a)) = ϵ_{D}(a mod \sqrt{D})a.

The theta-series

θ_{ψ} := \frac{1}{2} ∑_{a∈O_{K}} ψ(a)q^{aā} ∈ S_{2}(Γ_{0}(D^{2})) (q = e^{2πiτ})

is the weight two normalised eigenform of level D^{2} attached to A.

Fix an integer r ≥ 0, and consider the higher weight theta series

θ_{ψ^{r+1}} := \frac{1}{2} ∑_{a∈O_{K}} ψ(a)^{r+1}q^{aā} = ∑_{n=1}^{∞} a_{n}q^{n} ∈ \{ \begin{matrix}S_{r+2}(Γ_{0}(D^{2})) if r is even; \\ S_{r+2}(Γ_{0}(D), ϵ_{K}) if r is odd. \end{matrix}  (8)

Set

Γ = \{ \begin{matrix}Γ_{0}(D) if r is odd, \\ Γ_{0}(D^{2}) if r is even, \end{matrix}

and write C for the modular curve attached to Γ. Let W_{r} be the Kuga-Sato variety obtained by taking a canonical desingularisation of the r-fold fiber product

E ×_{C} E ×_{C} ⋯ ×_{C} E

of the universal (generalised) elliptic curve E over C. The locus W_{r}^{0} ⊂ W_{r} that lies over the open modular curve admits an explicit complex uniformisation

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

The theta series θ_{ψ^{r+1}} has a geometric interpretation as a regular (r+1)-form on W_{r} given on W_{r}^{0}(ℂ) by

ω_{ψ^{r+1}} = (2πi)^{r+1}θ_{ψ^{r+1}}(τ)dz_{1} ⋯ dz_{r}dτ,

where (z_{1}, …, z_{r}, τ) are the standard coordinates on ℂ^{r} × H. The q-expansion principle implies that

ω_{ψ^{r+1}} belongs to Ω^{r+1}(W_{r}/ℚ) = Fil^{r+1} H_{dR}^{r+1}(W_{r}/ℚ).

The Deligne-Scholl motive associated to ψ^{r+1} corresponds to the piece of the (r+1)-st cohomology of W_{r} on which the nth Hecke correspondence T_{n} acts (for each n) as multiplication by the Fourier coefficient a_{n} of (8). It can be shown that the étale realisations of this motive are isomorphic to a specific piece of the middle cohomology of A^{r+1}:

H_{et}^{r+1}(W̄_{r}, ℚ_{p})^{θ_{ψ^{r+1}}} = H_{et}^{r+1}(Ā^{r+1}, ℚ_{p})^{ψ^{r+1}}.

This isomorphism gives a non-trivial Tate cycle

Π_{p} ∈ H^{2r+2}(\overline{W_{r} × A^{r+1}}, ℚ_{p})(r+1)^{G_{ℚ}}.

The existence of this Tate cycle suggests the following conjecture which is the basis for the definition of Chow-Heegner points on A.

Conjecture 3.2. There is an algebraic cycle class Π^{?} ∈ CH^{r+1}(W_{r}×A^{r+1})(K)⊗ℚ satisfying

Π_{dR}^{?∗}([ω_{A}^{r+1}]) ∼ [ω_{ψ^{r+1}}],  Π_{dR}^{?∗}([ω_{A}^{j}ω̄_{A}^{r+1−j}]) = 0,  for all 1 ≤ j ≤ r,

where

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

is the map on deRham cohomology induced by Π^{?}, and the symbol ∼ denotes equality up to multiplication by a non-zero scalar of ℚ^{×}.

Notice that the putative cycle Π^{?} is also an element of CH^{r+1}(X_{r} × A), where X_{r} is the (2r+1)-dimensional variety

X_{r} := W_{r} × A^{r}.

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

Φ^{?} : CH^{r+1}(X_{r})_{0}⟶A

defined over K. It is not hard to see that Φ^{?} is non-trivial. More precisely, a direct calculation reveals that

Φ_{dR}^{?∗}(ω_{A}) = ω_{ψ^{r+1}} ∧ η_{A}^{r},  (9)

where η_{A} is a suitably normalised class in H_{dR}^{0,1}(A/ℂ). (The fact that A has complex multiplication implies that the class η_{A} can be chosen to belong to H_{dR}^{1}(A/K).)

Generalised Heegner cycles on X_{r}. The article [BDP1] introduces and studies a collection of null-homologous, r-dimensional algebraic cycles on X_{r}, referred to as generalised Heegner cycles. These cycles, which extend the notion of Heegner cycles on Kuga-Sato varieties considered in [Scho], [Ne] and [Zh], are indexed by isogenies φ : A⟶A′, and are defined over abelian extensions of K. The cycle Δ_{φ} attached to φ is essentially equal to the r-fold product of the graph of φ:

Δ_{φ} := ϵ_{r}(Graph(φ)^{r}) ⊂ (A × A′)^{r} = (A′)^{r} × A^{r} ⊂^{(∗)} W_{r} × A^{r} = X_{r},  (10)

where the inclusion (*) arises by embedding (A′)^{r} in W_{r} as a fiber for the natural projection W_{r}⟶C. The projector ϵ_{r} that appears in (10) is a suitable idempotent in the ring of algebraic correspondences on X_{r}, which has the effect of making the cycle Δ_{φ} homologically trivial.

It can be shown, by adapting an argument of Schoen [Scho], that the cycles Δ_{φ} generate a subgroup of CH^{r+1}(X_{r})_{0}(K^{ab}) of infinite rank. The conjectural map Φ_{K^{ab}}^{?} sends these generalised Heegner cycles to points in A(K^{ab}). The resulting collection

{Φ_{K^{ab}}^{?}(Δ_{φ})}_{φ:A⟶A′}  (11)

of Chow-Heegner points should generate an infinite rank subgroup of A(K^{ab}), and should give rise to an “Euler system” in the sense of Kolyvagin. In the classical situation where r = 0, the variety X_{r} is just a modular curve and the existence of Π^{?} follows from Faltings’ proof of the Tate conjecture for curves. When r ≥ 1, the very existence of the collection of Chow-Heegner points relies, ultimately, on producing the algebraic cycle Π^{?} unconditionally.

Complex calculations. Since this is a workshop about explicit methods, we hasten to point out that even when the modular parametrisation Φ^{?} cannot be shown to exist, it can still be computed efficiently in practice, by complex analytic means.

The numerical calculation of Φ^{?} rests on the complex Abel-Jacobi map

AJ_{X_{r}} : CH^{r+1}(X_{r})_{0}(ℂ)⟶\frac{Fil^{r+1} H_{dR}^{2r+1}(X_{r}/ℂ)^{dual}}{ImH_{2r+1}(X_{r}(ℂ), ℤ)}  (12)

of Griffiths and Weil, which is defined by the rule:

AJ_{X_{r}}(Δ)(ω) = ∫Δ̃ ω,  (for any (2r+1)-chain Δ̃ with ∂Δ̃ = Δ).  (13)

This is a natural generalisation of the usual Abel-Jacobi map for elliptic curves:

AJ_{A} : A(ℂ) = CH^{1}(A)_{0}(ℂ)⟶\frac{Ω^{1}(A/ℂ)^{dual}}{ImH_{1}(A(ℂ), ℤ)},  (14)

which one recovers from (12) after replacing X_{r} by A and setting r = 0. The image of the Chow-Heegner point Φ^{?}(Δ_{φ}) under the Abel-Jacobi map (14) is computed by noting that:

AJ_{A}(Φ^{?}(Δ_{φ}))(ω_{A}) = AJ_{X_{r}}(Δ_{φ})(Φ_{dR}^{?∗}(ω_{A})) = AJ_{X_{r}}(Δ_{φ})(ω_{ψ^{r+1}} ∧ η_{A}^{r}),  (15)

where the first equality follows from the functorial properties of the Abel-Jacobi maps, and the second follows from (9).

Let N = D or D^{2} (depending on whether r is odd or even) and let φ : A⟶A′ be an isogeny from A to some elliptic curve A′. Suppose that A′(ℂ) is described as A′ = ℂ/⟨1, τ⟩, and that (A′)^{r} is embedded in W_{r} as the fiber above the point of C corresponding to the pair (ℂ/⟨1, τ⟩, 1/N). Suppose also that

φ^{∗}(2πidz) = ω_{A},

where z is the standard coordinate on ℂ/⟨1, τ⟩ = A′(ℂ). The last expression in (15) can be calculated from the following proposition, which is established in [BDP1], Thm. 3.14:

Proposition 3.3. Let 0 ≤ j ≤ r be an integer. For a complex isogeny φ as above, modulo the appropriate period lattice,

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

Setting j = 0, we find that Φ_{ℂ}^{?} = Φ_{ℂ}, where

Φ_{ℂ} : CH^{r+1}(X_{r})_{0}(ℂ)⟶A(ℂ)

is given by the explicit formula

Φ_{ℂ}(Δ_{φ}) = \frac{2πi}{(τ − τ̄)^{r}} ∫_{i∞}^{τ}(z − τ̄)^{r}θ_{ψ^{r+1}}(z)dz (mod Λ_{A}),

for an appropriate period lattice Λ_{A} attached to the elliptic curve A. Conjecture 3.2 on the existence of the modular parametrisation Φ^{?} implies the following explicit algebraicity statement:

Conjecture 3.4. Let H be a subfield of K^{ab} and let Δ_{φ} ∈ CH^{r+1}(X_{r})_{0}(H) be a generalised Heegner cycle defined over H. Then (after fixing an an embedding of K^{ab} 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 3.2, Conjecture 3.4 has the virtue of being more readily amenable to experimental verification. A number of such verifications—which can be viewed as indirect numerical “tests” of the Tate conjectures for W_{r} × A^{r+1}—are documented in [BDP2]. In these experiments, the complex points Φ_{ℂ}(Δ_{φ}) attached to a few generalised Heegner cycles Δ_{φ} are calculated to high accuracy and recognized as algebraic points defined over the predicted class fields.

p-adic methods. Aside from such numerical explorations, the main theoretical evidence for the existence of the modular parametrisation Φ^{?} arises from p-adic methods.

If F is any field, then the Abel-Jacobi map admits an analogue in étale cohomology:

AJ_{F}^{et} : CH^{r+1}(X_{r})_{0}(F)⟶H^{1}(F, H_{et}^{2r+1}(X̄_{r}, ℚ_{p})(r+1)).  (16)

The image of the conjectural algebraic cycle Π^{?} under the étale cycle class map is a Tate cycle

Π_{et} ∈ H_{et}^{2r+2}(\overline{X_{r} × A}, ℚ_{p})(r+1)^{G_{ℚ}},

which in turn gives rise to a surjective, G_{ℚ}-equivariant projection

π_{r} : H_{et}^{2r+1}(X̄_{r}, ℚ_{p})(r+1)⟶H_{et}^{1}(Ā, ℚ_{p})(1) = V_{p}(A).

Applying π_{r} to the target of (16) gives a map

π_{r} ∘ AJ_{F}^{et} : CH^{r+1}(X_{r})_{0}(F)⟶H_{Sel}^{1}(F, V_{p}(A)),  (17)

where H_{Sel}^{1}(F, V_{p}(A)) is the pro-p Selmer group of A over F. This Selmer group consists cohomology classes whose restrictions to each completion F_{v} of F belongs to the image of the local connecting homomorphism

δ_{v} : (lim_{←} A(F_{v})/p^{n}A(F_{v})) ⊗_{ℤ_{p}} ℚ_{p}⟶H^{1}(F_{v}, V_{p}(A))

arising from the p-power descent exact sequence of Kummer theory for A over F_{v}. When F is a global field, this is the “usual” pro-p Selmer group of A over F, and when F is a local field of residue characteristic p, the ℚ_{p}-vector space H_{Sel}^{1}(F, V_{p}(A)) is identified with the Lie algebra A^{1}(F) ⊗ ℚ_{p} of the p-adic Lie group A(F).

Remark: The system {AJ_{F_{φ}}^{et}(Δ_{φ})}_{φ} (as φ ranges over all isogenies from A, and F_{φ} is the field of definition of φ) is an infinite collection of global cohomology classes defined over finite abelian extensions of K, satisfying various norm compatibility and Selmer conditions. This collection obeys (a simple variant of) the axioms of an Euler system, as they are spelled out in [Ru] for example.

In the case where F is a finite extension of ℚ_{p}, equation (17) can be used to define a p-adic parametrisation

Φ_{F} := π_{r} ∘ AJ_{F}^{et} : CH^{r+1}(X_{r})_{0}(F)⟶A(F) ⊗ ℚ,

which is a p-adic counterpart of the map Φ_{ℂ}, is defined independently of the Hodge or Tate conjectures, and agrees with Φ_{F}^{?} when the latter exists. The main theorem of [BDP2] is the following p-adic analogue of Conjecture 3.4, which shows that the images of generalised Heegner points under Φ_{F} have the expected algebraicity properties, and can be related to the L-series of A. Assume for simplicity that the integer r is odd.

Theorem 3.5. Let p = 𝔭𝔭̄ be a rational prime which splits in K/ℚ. Let H ⊂ K^{ab} be a finite extension of ℚ which is unramified at p, let Δ ∈ CH^{r+1}(X_{r})_{0}(H) be a generalised Heegner cycle defined over this field, and let H_{𝔭} ⊃ H be the completion of H at a prime above 𝔭. Then

Φ_{H_{𝔭}}(Δ) belongs to A(H) ⊗ ℚ.

In particular, the cycle Δ_{1} ∈ CH^{r+1}(X_{r})_{0}(K) attached to the identity isogeny 1 : A⟶A maps to a rational point on A(K) ⊗ ℚ under Φ_{K_{𝔭}}. This point is of infinite order if and only if

L(ψ^{2r+1}, r + 1) ≠ 0, and L′(ψ, 1) ≠ 0.

The idea of the proof of Theorem 3.5 is to express the local points Φ_{H_{𝔭}}(Δ_{φ}) in terms of special values of the p-adic L-functions studied in [BDP1] which are attached to the Rankin convolution of θ_{ψ^{r+1}} with Hecke characters of K. The resulting formulae for the local points Φ_{F} (Δ_{φ}) (for F any p-adic field over which Δ_{φ} can be defined) allows one to compare these points for different values of r, and thereby reduce the case r > 0 of Theorem 3.5 to the case r = 0, where it follows from the Tate conjecture for curves proved by Faltings.

The very possibility of such a proof reveals that the Chow-Heegner points constructed in this setting are not genuinely new, since they can ultimately be related to CM points on modular curves. The set-up involving the CM elliptic curve A and the variety X_{r}—a simple but non-trivial “toy model” for the notion of Chow-Heegner points—is perhaps most noteworthy for bringing the Hodge and Tate conjectures, which are notoriously difficult to test numerically, a bit closer to the realm of “explicit methods”.

4 ATR cycles

Of course, the hope is that higher-dimensional cycles will lead to points on E that cannot already be obtained by more classical approaches based on Heegner points. We will take a first step in this direction by considering certain non-algebraic cycles on Hilbert modular varieties.

The setting. Let F be a totally real field of degree r + 1, and fix an ordering v_{0}, v_{1}, …, v_{r} of the r+1 distinct real embeddings of F. Let E be an elliptic curve over F, and let

E_{j} := E ⊗_{v_{j}} ℝ (0 ≤ j ≤ r)

be the r+1 elliptic curves over ℝ obtained by taking the base change of E to ℝ via the embedding v_{j}. To ease the exposition, we will make the following inessential assumptions:

1. The field F has narrow class number one;

2. the conductor of E/F is equal to 1 (i.e., E has everywhere good reduction).

Remark 4.1. These hypotheses, although very restrictive, are satisfied in some examples. For example, when D = 29, 37 and 41, the real quadratic field F = ℚ(\sqrt{D}) has narrow class number one, and there is an elliptic curve E of conductor one over F. This elliptic curve cannot be defined over ℚ, but it is isogenous to its Galois conjugate, and is a quotient of the Jacobian J_{1}(D). The elliptic modular form thus associated to E belongs to S_{2}(Γ_{0}(D), ϵ_{D}), where ϵ_{D} is the quadratic Dirichlet character of conductor D attached to F.

In general, the modularity conjecture asserts that E gives rise to a Hilbert modular form f on SL_{2}(O_{F}). Such a form is a holomorphic function on the product H_{0} × H_{1} × ⋯ × H_{r} of r+1 copies of the complex upper half plane, which is of parallel weight (2, 2, …, 2) under the action of the Hilbert modular group SL_{2}(O_{F}). The latter group acts discretely on H_{0} × ⋯ × H_{r} by Möbius transformations via the embedding

(v_{0}, …, v_{r}) : SL_{2}(O_{F})⟶SL_{2}(ℝ)^{r+1}.

Because of this trasformation property, the Hilbert modular form f can be interpreted geometrically as a holomorphic differential (r+1)-form on the complex analytic quotient

X(ℂ) := SL_{2}(O_{F})\(H_{0} × H_{1} × ⋯ × H_{r}),  (18)

by setting

ω_{f}^{hol} := (2πi)^{r+1}f(τ_{0}, …, τ_{r})dτ_{0} ⋯ dτ_{r}.

This quotient in (18) is identified with the complex points of the (open) Hilbert modular variety X attached to GL(2)_{/F}, but this algebraic structure will not be exploited in our construction of Stark-Heegner points attached to ATR cycles.

It will be useful to replace ω_{f}^{hol} by a closed, but non-holomorphic differential (r+1)-form ω_{f} on X(ℂ). When r = 1, the differential ω_{f} is defined by choosing a unit ϵ ∈ O_{F}^{×} of norm −1 satisfying

ϵ_{0} := v_{0}(ϵ) > 0, ϵ_{1} := v_{1}(ϵ) < 0,

and setting

ω_{f} = (2πi)^{2} (f(τ_{0}, τ_{1})dτ_{0}dτ_{1} − f(ϵ_{0}τ_{0}, ϵ_{1}τ̄_{1})dτ_{0}dτ̄_{1}) .

For general r, one defines ω_{f} similarly, but this time summing over the subgroup of O_{F}^{×}/(O_{F}^{+})^{×} of cardinality 2^{r} consisting of units ϵ with v_{0}(ϵ) > 0. Note that the closed (r+1)-form ω_{f} is holomorphic in τ_{0}, but only harmonic in the remaining variables τ_{1}, …, τ_{r}. The justification for working with ω_{f} rather than ω_{f}^{hol} lies in the following statement which is a reformulation of a conjecture of Oda [Oda].

Conjecture 4.2 (Oda). Let

Λ_{f} := {∫_{γ} ω_{f}, γ ∈ H_{r}(X(ℂ), ℤ)} .

Then Λ_{f} is a lattice in ℂ and the elliptic curve ℂ/Λ_{f} is isogenous to E_{0}.

This conjecture is known to hold for Hilbert modular forms which are base change lifts of classical elliptic modular forms. For example, in the setting of Remark 4.1, the Hilbert modular form attached to E is the Doi-Naganuma lift of an elliptic modular form in S_{2}(Γ_{1}(D), ϵ_{D}) and Conjecture 4.2 is known to hold in this case.

Let

Z_{r}(X(ℂ)) := \{ \begin{matrix}Null-homologous cycles \\ of real dimension r \\ on X(ℂ) \end{matrix} \} .

Conjecture 4.2 makes it possible to define an “Abel-Jacobi map”

AJ_{f} : Z_{r}(X(ℂ))⟶E_{0}(ℂ),  (19)

by choosing an isogeny ι : ℂ/Λ_{f}⟶E_{0}(ℂ), and setting

AJ_{f}(Δ) := ι (∫Δ̃ ω_{f}) ,  (for any Δ̃ with ∂Δ̃ = Δ).  (20)

Note that the domain Z_{r}(X(ℂ)) of AJ_{f} has no natural algebraic structure, and that the map AJ_{f} bears no obvious relation (beyond an analogy in its definition) with the Griffiths-Weil Abel-Jacobi map on the Hilbert modular variety X.

ATR Cycles. A quadratic extension K of F is called an ATR extension if

K ⊗_{F,v_{0}} ℝ ≃ ℂ, K ⊗_{F,v_{j}} ℝ ≃ ℝ ⊕ ℝ, (1 ≤ j ≤ r).

The acronym ATR stands for “Almost Totally Real”; an ATR extension of F is “as far as possible” from being a CM extension, without being totally real.

Fix an ATR extension K of F, and let Ψ : K⟶M_{2}(F) be an F -algebra embedding. Then

1. Since K ⊗_{F,v_{0}} ℝ ≃ ℂ, the torus Ψ(K^{×}) has a unique fixed point τ_{0} ∈ H_{0}.

2. For each 1 ≤ j ≤ r, the fact that K ⊗_{F,v_{j}} ℝ ≃ ℝ ⊕ ℝ shows that Ψ(K^{×}) has two fixed points τ_{j} and τ′_{j} on the boundary of H_{j}. Let Υ_{j} ⊂ H_{j} be the hyperbolic geodesic joining τ_{j} to τ′_{j}.

An embedding Ψ : K⟶M_{2}(F) has a conductor, which is defined to be the unique O_{F} -ideal c_{Ψ} for which

Ψ(K) ∩ M_{2}(O_{F}) = Ψ(O_{F} + c_{Ψ}O_{K}).

The O_{F} -order O_{Ψ} := O_{F} + c_{Ψ}O_{K} is called the order associated to Ψ. By the Dirichlet unit theorem, the group

Γ_{Ψ} := Ψ((O_{Ψ}^{+})^{×}) ⊂ SL_{2}(O_{F})

is of rank r and preserves the region

R_{Ψ} := {τ_{0}} × Υ_{1} × ⋯ × Υ_{r}.

The ATR cycle associated to the embedding Ψ is defined to be the quotient

Δ_{Ψ} := Γ_{Ψ}\R_{Ψ}.

It is a closed cycle on X(ℂ) which is topologically isomorphic to an r-dimensional real torus. In many cases, one can show that Δ_{Ψ} is null-homologous, at least after tensoring with ℚ. (This is the case, for instance, when r = 1, and it follows from the fact that the group cohomology H^{r}(SL_{2}(O_{F}), ℂ) is trivial.) Assume from now on that Δ_{Ψ} is homologically trivial, and therefore that it belongs to Z_{r}(X(ℂ)).

The following conjecture lends arithmetic meaning to the Abel-Jacobi map AJ_{f} and to the ATR cycles Δ_{Ψ}.

Conjecture 4.3. Let Ψ : K⟶M_{2}(F) be an F -algebra embedding of an ATR extension K of F. Then the complex point AJ_{f}(Δ_{Ψ}) ∈ E_{0}(ℂ) is algebraic. More precisely, the isogeny ι in the definition (20) of AJ_{f} can be chosen so that, for all Ψ,

AJ_{f}(Δ_{Ψ}) belongs to E(H_{c_{Ψ}}),

where H_{c_{Ψ}} is the ring class field of K of conductor c_{Ψ}.

This conjecture has been tested numerically in [DL], for the three elliptic curves mentionned in Remark 4.1. A key ingredient in [DL] is the formulation of an efficient algorithm for calculating AJ_{f} numerically. This algorithm relies on group cohomology, and involves the manipulation of certain (r+1)-cochains on Γ which are defined by integrating ω_{f} over appropriate regions. The algorithm described in [DL] also exploits the fact that the real quadratic field K = ℚ(\sqrt{D}) for D = 29, 37, and 41, is Euclidean. It would be of interest to have algorithms to calculate AJ_{f} in more general settings, particularly in cases where r > 1.

Conjecture 4.3 is poorly understood at present. For instance, it is not clear whether the Tate conjecture sheds any light on it. On the positive side, the ATR points that are produced by Conjecture 4.3 are “genuinely new” and go beyond what can be obtained using only CM points on Shimura curves. Indeed, the former are defined over abelian extensions of ATR extensions of totally real fields, while the latter are defined over abelian extensions of CM fields.

5 Real quadratic cycles on SL_{2}(ℤ[1/p])\(H_{p} × H)

The construction based on ATR cycles fails to cover some of the most basic settings where a modular construction might be expected to exist. The simplest non-trivial such setting arises when E is an elliptic curve over ℚ of prime conductor p, and K is a real quadratic field in which p is inert. In that case, a study of signs in functional equations reveals that

ord_{s=1} L(E/H, s) ≥ [H : K],

for any abelian extension H of K which is unramified at p and for which Gal(H/K) is isomorphic to a (generalised) dihedral group. (See the discussion in the introduction of [Da01] for example.) The Birch and Swinnerton-Dyer conjecture therefore predicts that

rank(E(H)) ≥^{?} [H : K].

It is natural to ask whether this predicted systematic growth in Mordell-Weil rank can be accounted for by a modular construction.

Such a modular construction does appear to exist. It rests on the formal analogy between the Hilbert modular surface SL_{2}(O_{F})\(H_{0} × H_{1}) (corresponding to the case r = 1 of the ATR construction described in the previous paragraph) and the quotient

SL_{2}(ℤ[1/p])\(H_{p} × H),

where H_{p} := P_{1}(ℂ_{p}) − P_{1}(ℚ_{p}) is the p-adic upper half plane. Some of the terms that make up the analogy are listed in the table below.

ATR cycles  Real quadratic cycles

F real quadratic ℚ

v_{0}, v_{1}  p, ∞

Elliptic curve E/F of conductor 1

Elliptic curve E/ℚ of conductor p

SL_{2}(O_{F})\(H_{0} × H_{1})  SL_{2}(ℤ[1/p])\(H_{p} × H)
K/F ATR  K/ℚ real quadratic, with p inert
Ψ : K⟶M_{2}(F)  Ψ : K⟶M_{2}(ℚ)
⟨γ⟩ := Ψ((O_{Ψ}^{+})^{×})  ⟨γ⟩ := Ψ((O_{Ψ}^{+})^{×})
Δ_{Ψ} = {τ_{0}} × (Υ_{1}/γ), τ ∈ H_{0}  Δ_{Ψ} = {τ} × (Υ_{1}/γ), τ ∈ H_{p}.
⇓ AJ_{f}  ⇓ AJ_{f}^{(p)}
Points in ℂ/Λ_{f} = E_{0}(ℂ),  Points in K_{p}^{×}/q^{ℤ} = E(K_{p}),

defined over abelian extensions of K defined over abelian extensions of K.

The “real quadratic cycles” Δ_{Ψ} in SL_{2}(ℤ[1/p])\(H_{p} × H) are topologically isomorphic to ℝ/ℤ, and AJ_{f}^{(p)}(Δ_{Ψ}) belongs to K_{p}^{×}/q^{ℤ} = E(K_{p}), where q ∈ ℚ_{p}^{×} is the p-adic Tate period of E. Since the symmetric space H_{p} ×H mixes a rigid analytic topology on the first factor with a complex analytic topology on the second, one cannot define AJ_{f}^{(p)} by directly integrating an appropriate differential on a two-dimensional region having Δ_{Ψ} as boundary, as in equation (20) defining AJ_{f}. The main steps that make it possible to define the p-adic analogue of AJ_{f} are:

1. To reinterpret the elliptic modular form f ∈ S_{2}(Γ_{0}(p)) attached to E as a “mock Hilbert modular form” on SL_{2}(ℤ[1/p])\H_{p} × H. This reinterpretation gives a precise meaning to certain 2-cochains on Γ with values in ℂ_{p}^{×} which are the direct p-adic analogues of the corresponding cochains considered in the AT R setting in the algorithms of [DL].

2. With these cochains in hand, the algorithms of [DL] can be precisely mimicked, yielding invariants AJ_{f}^{(p)}(Δ_{Ψ}) ∈ K_{p}^{×}/q^{ℤ}.

For more details on this construction, and the precise definition of AJ_{f}^{(p)}, see [Da01], [Da04]. The article [DP] describes the most efficient algorithms for computing the Stark-Heegner points AJ_{f}^{(p)}(Δ_{Ψ}) attached to real quadratic fields. These algorithms have been implemented in MAGMA and can be downloaded from the web site

http://www.math.mcgill.ca/darmon/programs/shp/shp.html

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