Heegner points, Heegner cycles, and congruences

Henri Darmon

May 23, 2000

Contents

1 Heegner objects 2 1.1 Modular elliptic curves . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Binary quadratic forms . . . . . . . . . . . . . . . . . . . . . . 4 1.3 The case D < 0: Heegner points . . . . . . . . . . . . . . . . . 6 1.4 The case D > 0: Heegner cycles . . . . . . . . . . . . . . . . . 7 1.5 Properties of the Heegner objects . . . . . . . . . . . . . . . . 8

1.5.1 Action of the Hecke operators and w_{N} . . . . . . . . . 8 1.5.2 Behaviour under norms . . . . . . . . . . . . . . . . . . 9

2 Relation with L-functions 10 2.1 Root numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Formulas of Gross-Zagier and Waldspurger . . . . . . . . . . . 11

3 A refined conjecture 12 3.1 Motivation and statement . . . . . . . . . . . . . . . . . . . . 12 3.2 Properties of θ_{D} . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.3 The leading coefficient . . . . . . . . . . . . . . . . . . . . . . 15 3.4 The case D < 0: theoretical evidence . . . . . . . . . . . . . . 16 3.5 The case D > 0: computational evidence . . . . . . . . . . . . 17

3.5.1 The curve X_{0}(11) . . . . . . . . . . . . . . . . . . . . . 17 3.5.2 The curve X_{0}(37)^{+} . . . . . . . . . . . . . . . . . . . . 19 3.5.3 The curve of conductor 5077 . . . . . . . . . . . . . . . 23

1This research was funded in part by an NSERC postdoctoral fellowship.

Abstract We define certain objects associated to a modular elliptic curve E and a discriminant D satisfying suitable conditions. These objects interpolate special values of the complex L-functions associated to E over the quadratic field Q(\sqrt{D}), in the same way that Bernouilli numbers interpolate special values of Dirichlet L-series. Following an approach of Mazur and Tate [MT], one can make conjectures about congruences satisfied by these objects which are resonant with the usual Birch and Swinnerton-Dyer conjectures. These conjectures exhibit some surprising features not apparent in the classical case.

1 Heegner objects

1.1 Modular elliptic curves

Let E be an elliptic curve defined over Q by the Weierstrass equation

y^{2} = 4x^{3} − g_{2}x − g_{3}, g_{2}, g_{3} ∈ Z,

and let N denote the arithmetic conductor of E, which can be computed from g_{2} and g_{3} by Tate’s algorithm [Ta]. To simplify the discussion, let us assume that N is odd.

The group E(C) is isomorphic to the complex torus C/Λ, where Λ is a free Z-lattice of rank 2. The lattice Λ can be computed explicitely from the coefficients g_{2} and g_{3}, by using the arithmetic-geometric mean. An isomorphism from C/Λ to E is given by

z ↦ (℘(z), ℘′(z)),

where ℘(z) denotes the Weierstrass ℘-function.

The set E_{ns}(F_{p}) of non-singular points on the reduction of E mod p is a finite abelian group. Let N_{p} denote its order, and define a_{p} = p + δ_{p} − N_{p}, where δ_{p} = 0 if E has bad reduction at p, and is equal to 1 otherwise. Extend this to a_{n}, for any positive integer n, by equating coefficients in the formal series identity:

∑_{n≥0} a_{n}n^{−s} = ∏_{p}(1 − a_{p}p^{−s} + δ_{p}p^{1−2s})^{−1}.

¿From the inequality of Hasse, |a_{p}| ≤ 2\sqrt{p}, it follows that the Fourier series

f_{E}(τ) = ∑_{n=1}^{∞} a_{n}e^{2πinτ}

converges absolutely for im(τ) > 0. Hence the differential ω_{E} = 2πif_{E}(τ)dτ is a holomorphic differential on the upper half plane H. The group SL_{2}(R) of matrices of determinant 1 acts (on the left) on H by

( \begin{matrix}a & b \\ c & d \end{matrix} ) τ = \frac{aτ + b}{cτ + d}.

Taking pullbacks defines a right action of SL_{2}(R) on the space of holomorphic functions or differentials on H.

The Atkin-Lehner involution w_{N} and the Hecke operators T_{p} act on the space of differentials by

T_{p}g(τ)dτ = pg(pτ)dτ + 1/p ∑_{k=0}^{p−1} g(\frac{τ + k}{p})dτ,

w_{N}g(τ)dτ = g(−1/Nτ)d(−1/Nτ).

Let Γ_{0}(N) denote the group of matrices in SL_{2}(Z) whose lower left entry is divisible by N. The following conjecture is supported by extensive computational and theoretical evidence, and is widely believed to be true:

Conjecture 1.1 (Shimura-Taniyama-Weil) .

1. The differential ω_{E} is invariant under the action of Γ_{0}(N), i.e.,

f_{E}(\frac{aτ + b}{cNτ + d}) = (cNτ + d)^{2}f_{E}(τ),

for all ( \begin{matrix}a & b \\ cN & d \end{matrix} ) of determinant 1 with a, b, c, d ∈ Z.

2. The set L of all ∫_{z}^{γz} ω_{E}, with γ ∈ Γ_{0}(N) and z ∈ H, is a lattice in C, and there exists an integer λ such that λL ⊂ Λ. (Hence the map ϕ : τ ↦ λ ∫_{i∞}^{τ} ω_{E} is a surjective analytic map from H/Γ_{0}(N) to C/Λ.)

3. The differential ω_{E} is an eigenvalue for the Hecke operators T_{p} and the Atkin-Lehner involution,

T_{p}ω_{E} = a_{p}ω_{E}, (p, N) = 1, w_{N}ω_{E} = ϵω_{E}, ϵ = ±1.

We assume that this conjecture is true for E - in theory, it can be checked by a finite amount of computation, cf. [Me]. All of our constructions rely crucially on the truth of the Shimura-Taniyama-Weil conjecture for E.

1.2 Binary quadratic forms

Let D = D_{0}f^{2}, where D_{0} is a fundamental discriminant and f is a square-free integer prime to D_{0}, and let K = Q(\sqrt{D}) = Q(\sqrt{D_{0}}) be the corresponding quadratic field with ring of integers O_{K}. Let O_{f} be the order in O_{K} of conductor f, consisting of all elements in O_{K} which are congruent to a rational integer modulo f.

Definition 1.2 The pair (E, D) satisfies the Heegner hypothesis if for all p dividing N, the Kronecker symbol (\frac{D}{p}) is equal to 1.

Assume from now on that (E, D) satisfies the Heegner hypothesis; then there is an integer B_{0} such that

B_{0}^{2} ≡ D_{0} (mod 4N).

Fix such a B_{0} once and for all. The standard shorthand (A, B, C) will be used to denote the binary quadratic form Ax^{2} + Bxy + Cy^{2}.

Definition 1.3 A quadratic form F = (A, B, C) is said to be Heegner if N divides the coefficient A, and B ≡ B_{0}f (mod 2N).

Let F denote the set of primitive binary quadratic forms of discriminant D, and let F_{N} denote the set of primitive binary quadratic forms which are Heegner. The group SL_{2}(Z) acts on the right on F by the rule:

F (x, y) ( \begin{matrix}a & b \\ c & d \end{matrix} ) = F (ax + by, cx + dy).

Proposition 1.4 .

1. The set F_{N} is stable under the action of Γ_{0}(N).

2. The natural map F_{N}/Γ_{0}(N) ⟶ F/SL_{2}(Z) is an isomorphism.

To show surjectivity in 2, let F be a form in F, and suppose without loss of generality that F = (A, B, C) with (C, N) = 1. (One can bring F to this form by modifying it by an element of SL_{2}(Z).) Let t be an integer satisfying the congruence

tC ≡ \frac{B_{0}f − B}{2} (mod N),

and let γ = ( \begin{matrix}1 & 0 \\ t & 1 \end{matrix} ). Then F γ is an SL_{2}(Z)-representative of F which

belongs to F_{N}. To show injectivity, one may proceed by a direct argument. This part of the proof breaks down when D is not prime to N. (In our case, (D, N) = 1 follows from the Heegner hypothesis.) A more general statement which also covers the cases where D and N are not assumed to be coprime is given in [GKZ], pp. 504-506.

Let h^{+}(D) denote the number of inequivalent primitive binary quadratic forms of discriminant D. Proposition 1.4 says that the set F_{N}/Γ_{0}(N) is finite and has order h^{+}(D). In addition, F_{N}/Γ_{0}(N) inherits a group structure from the Gaussian composition of binary quadratic forms in F/SL_{2}(Z). Let G_{D} denote the set F_{N}/Γ_{0}(N) endowed with this group structure.

Class field theory interprets G_{D} as the Galois group of an abelian extension of K:

G_{D} = Gal(K_{f}/K),

where K_{f} is the ring class field of K associated to the order of conductor f (cf. [Co], pp. 180-182). The following proposition gives the order of G_{D} in terms of h, the class number of the field K.

Proposition 1.5 .

1. If D < 0, then

h^{+}(D) = hu^{−1} ∏_{p|f}(p − (\frac{p}{D_{0}} )),

where u = #O_{K}^{∗}/2.

2. If D > 0, let w be a fundamental unit of K, and let u be the smallest integer such that w^{u} is congruent to a rational integer modulo f, and w^{u} is totally positive. Then

h^{+}(D) = hu^{−1}2 ∏_{p|f}(p − (\frac{p}{D_{0}} )).

Our program in the next two sections is to associate to each (not necessarily primitive) Heegner form F = (A, B, C) of discriminant D a Heegner object which belongs to a certain Z-module M_{D}.

The nature of the Heegner construction, and the nature of M_{D}, depends on whether D is positive or negative; we will treat those two cases separately.

1.3 The case D < 0: Heegner points

If D < 0, the two roots of the dehomogenized form Ax^{2}+Bx+C are complex conjugate and distinct, and there is a unique root τ_{0} which lies in the upper half plane. Let

q_{0} = e^{2πiτ_{0}},

and let

z_{0} = ∑_{n=1}^{∞} \frac{a_{n}}{n} q_{0}^{n}.

Theorem 1.6 The value of z_{0} is independent of the choice of the Γ_{0}(N)- representative of F, up to addition of elements of Λ ⊂ C.

Proof: Since the root of F γ in the upper-half plane is γ^{−1}τ_{0}, the value of τ_{0} is well defined in H/Γ_{0}(N). But z_{0} = ϕ(τ_{0}), where ϕ is the map defined in conjecture 1.1. Hence theorem 1.6 follows from conjecture 1.1 which we assumed to be true in our case.

By theorem 1.6, the point α_{F} = (x_{0}, y_{0}) = (℘(z_{0}), ℘′(z_{0})) is a well-defined complex point on E corresponding to F ∈ G_{D}, and satisfying the Weierstrass equation

y_{0}^{2} = 4x_{0}^{3} − g_{2}x_{0} − g_{3}.

Theorem 1.7 The complex numbers x_{0} and y_{0} satisfy an algebraic equation of degree 2h^{+}(D) over Q, and can be viewed as elements of the ring class field K_{f}. Hence α_{F} belongs to the Mordell-Weil group E(K_{f}).

This is a consequence of the theory of complex multiplication. See [Cx], chapter three, or [Sh].

Let M_{D} = E(K_{f}). It is a finitely generated Z-module by the Mordell-Weil theorem.

1.4 The case D > 0: Heegner cycles

When D > 0, the roots τ_{1} and τ_{2} of the dehomogenized form Ax^{2} + Bx + C lie on the real line. Let C be the geodesic in the hyperbolic plane joining τ_{1} and τ_{2}. A geodesic joining two rational numbers on the real line maps to a cycle of finite length on X_{0}(N), joining a cusp to a cusp. In general, an arbitrary geodesic joining irrational real numbers maps to a path with dense image on X_{0}(N). Because τ_{1} and τ_{2} are real quadratic and conjugate, the image of C on X_{0}(N)(C) is an infinite periodic cycle. More precisely, the form Ax^{2} + Bxy + Cy^{2} is preserved by an infinite abelian subgroup of SL_{2}(Z) of rank 1. A generator for this group modulo torsion is called an automorph of F. When F is primitive, the automorph M_{F} can be written down by choosing a fundamental solution to Pell’s equation

u^{2} − Dv^{2} = 1,

and setting

M_{F} = ( \begin{matrix}u − Bv & −2Cv \\ 2Av & u + Bv \end{matrix} ) .

Observe that M_{F} belongs to Γ_{0}(N), since N divides A. Now choose any point τ lying on the geodesic C, and let C_{i} be the geodesic joining M^{i}τ to M^{i+1}τ. Then C can be expressed as an infinite union

C = ∪_{i∈Z}C_{i},

and each C_{i} maps to the same basic homology cycle α_{F} in M_{D} = H_{1}(E(C), Z). The construction of α_{F} also works when F is not primitive, using the same definiton for M_{F}; but observe that in this case the matrix M_{F} could be a non-trivial power of the automorph of F.

The module M_{D} can be identified with the lattice Λ by the map α ↦ ∫_{α} ω_{E}. Thus α_{F} can be computed by evaluating the integral

∫_{z}^{M_{F} z} ω_{E}

which belongs to Λ and does not depend on the choice of z, and using the identification of Λ with H_{1}(E(C), Z).

Theorem 1.8 Once a choice of a fundamental solution to Pell’s equation has been made, the cycle α_{F} does not depend on the choice of F modulo the action of Γ_{0}(N), and hence the assignment F ↦ α_{F} is well-defined on G_{D}.

Proof: Modifying F by an element γ in Γ_{0}(N) has the effect of conjugating M_{F} by γ, i.e.,

M_{F γ} = γ^{−1}M_{F} γ.

But the map ψ : M ↦ ∫_{z}^{Mz} ω_{E} is a homomorphism from Γ_{0}(N) to the abelian group Λ, and hence ψ(M_{F γ}) = ψ(M_{F}).

1.5 Properties of the Heegner objects

1.5.1 Action of the Hecke operators and w_{N}

Let F ∈ G_{D} be a class of binary quadratic forms represented by the Heegner form (A, B, C). The Hecke operators can be defined on the forms by the rule

T_{p}F = F_{∞} + ∑_{k=0}^{p−1} F_{k},

F_{∞} = (A, Bp, Cp^{2}), F_{k} = (Ap^{2}, (B + 2Ak)p, Ak^{2} + Bk + C),

where the sum is to be viewed as a formal sum of (not necessarily primitive) binary quadratic forms of dicriminant Dp^{2}. Observe that the forms F_{∞} and F_{k} are Heegner forms, and hence the construction of sections 1.3 or 1.4 can be applied to them. Thus one defines

α_{T_{p}F} = α_{F_{∞}} + ∑_{k=0}^{p−1} α_{F_{k}}.

Proposition 1.9 .

1. If D < 0, then α_{T_{p}F} = a_{p}uα_{F}, where u = #O_{f}^{∗}/2.

2. If D > 0, let u be the smallest integer such that (v+w\sqrt{D})^{u} is congruent to a rational integer mod p, where (v, w) is a fundamental solution to the Pell equation v^{2} − Dw^{2} = 1. Then α_{T_{p}F} = a_{p}uα_{F}.

The Atkin-Lehner involution w_{N} assigns to the Heegner form F the Heegner form (CN, B, A/N).

Proposition 1.10 α_{w_{N} F} = −ϵα_{F}.

Propositions 1.9 and 1.10 can be verified by a direct argument, using the definition of the objects α_{F} in terms of ω_{E}, and the the action of the Hecke operators and the Atkin-Lehner involution on ω_{E} described in part 3 of conjecture 1.1. For more details (when D < 0) see [Gr1], §5 and §6, or [Gr2].

Let Frob_{N} be the class of quadratic forms in G_{D} represented by the form (N, B_{0}f, (B_{0}^{2}f^{2} − D)/(4N)). This terminology is appropriate, because this form correponds to the Frobenius element at P_{1} ⋯ P_{k}, where P_{j} is prime ideal above p_{j}|N given by (p_{j}, (B_{0} − \sqrt{D_{0}})/2). The action of the Atkinlehner involution w_{N} can be written down in terms of the form Frob_{N} and the Gaussian composition law, as:

Proposition 1.11 w_{N}F = Frob_{N}F^{−1}.

1.5.2 Behaviour under norms

Let p be a prime which does not divide ND, and let F = (A, B, C) be an element of G_{Dp^{2}}. Since the ring class field K_{f} is contained in the ring class field K_{fp}, there is a natural homomorphism μ_{p} : G_{Dp^{2}} ⟶ G_{D}. Let F̄ denote a form in G_{D} which represents μ_{p}(F). Define the norm of F to be the formal sum

N_{p}F = ∑_{μ_{p}(G)=F̄} G

of forms in G_{Dp^{2}}, and define the norm of α_{F} to be the element

N_{p}α_{F} = ∑_{μ_{p}(G)=F̄} α_{G},

where the sum is taken over all inequivalent primitive Heegner forms of discriminant Dp^{2} which map to F̄.

If p is split in K, choose an ideal P of K above p, and let Frob_{p} be the quadratic form in G_{D} which represents the frobenius element at P in Gal(K_{f}/K). Choosing a different prime of K above p replaces Frob_{p} by Frob_{p}^{−1}, so the element Frob_{p}+Frob_{p}^{−1} in the group ring Z[G_{D}] is well-defined.

Proposition 1.12 .

1. If (\frac{p}{D_{0}} ) = 1, then N_{p}α_{F} = a_{p}αF̄ − α_{Frob}_{p}F̄ − α_{Frob}_{p}^{−1}F̄.

2. If (\frac{p}{D_{0}} ) = −1, then N_{p}α_{F} = a_{p}αF̄.

Proof: The formal sum N_{p}F can be expressed in terms of the Hecke operators T_{p} by

N_{p}F = u^{−1}(T_{p}F̄) − Frob_{p}F̄ − Frob_{p}^{−1}F̄ if (\frac{p}{D_{0}} ) = 1,

and

N_{p}F = u^{−1}(T_{p}F̄) if (\frac{p}{D_{0}} ) = −1,

where u is as defined in proposition 1.9; the result follows from this proposition.

2 Relation with L-functions

2.1 Root numbers

Let L(E/K, s) denote the L-function of E over K, defined by the Euler product expansion

∏_{v}(1 − a_{Nv}Nv^{−s} + Nv^{1−2s})^{−1},

where the product is taken over the places v of K. This factors as a product of two L functions

L(E/K, s) = L(E/Q, s)L(E^{(D)}/Q, s),

where E^{(D)} is the twist of E by D defined by the equation

Dy^{2} = 4x^{3} − g_{2}x − g_{3}.

Let r_{an} denote the order of vanishing of L(E/K, s) at s = 1, and let r_{an}^{+} (resp. r_{an}^{−}) denote the order of vanishing of L(E/Q, s) (resp. L(E^{(D)}/Q, s)) at s = 1.

Proposition 2.1 .

1. If D < 0, then r_{an}^{+} ≢ r_{an}^{−} (mod 2), and r_{an} is odd.

2. If D > 0, then r_{an}^{+} ≡ r_{an}^{−} (mod 2), and r_{an} is even.

Proof: Let ϵ be the eigenvalue for the Atkin-Lehner involution w_{N} acting on ω_{E}. The sign in the functional equation for L(E/Q, s) is −ϵ, and hence L(E/Q, s) vanishes to odd order at the critical point s = 1 if ϵ = 1, and to even order if ϵ = −1. As explained in [Gr1], the sign in the functional equation for the L function of the twist E^{(D)} can be computed explicitely, and is equal to ϵ if D < 0, and −ϵ if D > 0, when D satisfies the Heegner hypothesis. The result follows.

2.2 Formulas of Gross-Zagier and Waldspurger

We define a Hermitian pairing ⟨ , ⟩_{D} on M_{D} ⊗ C.

Definition 2.2 The pairing ⟨ , ⟩_{D} : M_{D} × M_{D} ⟶ C is defined by:

1. If D < 0, let it be the Néron-Tate canonical height on E(K_{f}), extended to a Hermitian pairing on E(K_{f}) ⊗ C.

2. If D > 0, let ⟨α_{1}, α_{2}⟩_{D} = ∫_{α_{1}} ω_{E} ∫_{α_{2}} ω̄_{E}.

Given a complex character χ : G_{D} ⟶ C^{∗}, let

α_{χ} = \frac{1}{h^{+}(D)} ∑_{F ∈G_{D}} χ(F)α_{F} ∈ M_{D} ⊗ C.

To such a χ one can associate the twisted L-function

L(E/K, χ, s) = ∏_{v}(1 − χ(v)a_{Nv}Nv^{−s} + χ^{2}(v)Nv^{1−2s})^{−1}.

It has an analytic continuation to the complex plane. Let ω^{+}, ω^{−} ∈ Λ be the real and imaginary periods attached to E; the lattice generated by ω^{+} and ω^{−} is of index 1 or 2 in Λ.

The following formula gives the relation between the elements α_{F} and the special values L(E/K, χ, 1). For D < 0 it is a result of Gross and Zagier, and for D > 0, it was proved by Waldspurger.

Theorem 2.3 (Gross-Zagier, Waldspurger) Suppose that D = D_{0} is a fundamental discriminant, and let h = h^{+}(D).

1. If D < 0, then

⟨α_{χ}, αχ̄⟩_{D} =^{·} u^{2}\sqrt{D}h^{−1}(ω^{+}ω^{−})^{−1}L′(E/K, χ, 1),

where u = #O_{K}^{∗}/2.

2. If D > 0, then

⟨α_{χ}, αχ̄⟩_{D} =^{·} \sqrt{D}h^{−1}L(E/K, χ, 1),

where the symbol =^{·} denotes equality up to multiplication by a power of 2 which could be explicitly determined.

For the case D < 0, see [GZ], and for D > 0, see [GKZ], p. 527 and [Wal].

A similar formula certainly holds for non-fundamental D, but it has not been worked out for D < 0; one would need to adjust the formula to take into account the Euler factors at the primes dividing f. Since the formulas are given only to provide motivation for the later conjectures and results, we have not strived for the greatest generality and precision in writing them down.

3 A refined conjecture

3.1 Motivation and statement

Consider the formal elements

θ_{D} = ∑_{F ∈G_{D}} α_{F} · F, θ_{D}^{∗} = ∑_{F ∈G_{D}} α_{F} · F^{−1},

viewed as elements of the tensor product M_{D} ⊗ Z[G_{D}]. Let L_{D} be defined by taking the formal product:

L_{D} = θ_{D} · θ_{D}^{∗} ∈ M_{D}^{⊗2} ⊗ Z[G_{D}].

The pairing ⟨ , ⟩_{D} of section 2.2 is a linear map of M_{D}^{⊗2} to C and hence extends by linearity to a map from M_{D}^{⊗2} ⊗ Z[G_{D}] to C[G_{D}]. Applying this map to L_{D} gives an element L_{D}^{C} of C[G], which interpolates special values of the L-function of E over K; for the expression ⟨α_{χ}, αχ̄⟩_{D} which appears in theorem 2.3 is equal to χ(L_{D}^{C}). The original element L_{D} has more structure than L_{D}^{C}, since its coefficients are elements of a Z-module M_{D}^{⊗2}. For instance, it makes sense to talk about congruences for these coefficients. Let I denote the augmentation ideal in the integral group ring Z[G_{D}]. The first conjecture we make is close in spirit to the Birch and Swinnerton-Dyer conjecture, (and, even more so, to its p-adic avatars) and is inspired by a similar conjecture of Mazur and Tate in the cyclotomic case (cf. [MT], [D3]).

Conjecture 3.1 .

1. If D < 0, then L_{D} belongs to the subgroup M_{D}^{⊗2} ⊗I^{r−1} of M_{D}^{⊗2} ⊗Z[G_{D}].

2. If D > 0, then L_{D} belongs to the subgroup M_{D}^{⊗2} ⊗ I^{r} of M_{D}^{⊗2} ⊗ Z[G_{D}].

Let r denote the rank of E(K). Let E(K)^{+} and E(K)^{−} denote the plus and minus eigenspaces of E(K) under the action of the involution in Gal(K/Q). They generate a submodule of E(K) of index at most 2. Let r^{+} and r^{−} denote the ranks E(K)^{+} and E(K)^{−}; thus r^{+} is the rank of E over Q, and r^{+} +r^{−} = r. The Birch and Swinnerton-Dyer conjecture implies that r^{±} = r_{an}^{±}. Let ρ = max(r^{+}, r^{−}).

Guided by the main result of [D1], one is lead to make the following stronger conjecture about the “square root” θ_{D} of L_{D}.

Conjecture 3.2 .

1. If D < 0, then θ_{D} belongs to the subgroup M_{D} ⊗ I^{ρ−1} of M_{D} ⊗ Z[G_{D}].

2. If D > 0, then θ_{D} belongs to the subgroup M_{D} ⊗ I^{ρ} of M_{D} ⊗ Z[G_{D}].

When D < 0, conjecture 3.2 implies that L_{D} belongs to M_{D}^{⊗2}⊗I^{2ρ−2}. Assuming that r is odd (which is implied by the Birch Swinnerton-Dyer conjecture r = r_{an} together with proposition 2.1) one has 2ρ − 2 ≥ r − 1, with equality holding if and only if |r^{+} − r^{−}| = 1. When D > 0, conjecture 3.2 implies that L_{D} belongs to M_{D}^{⊗2} ⊗ I^{2ρ}, and 2ρ ≥ r, with equality holding if and only if r^{+} = r^{−}.

The evidence for conjecture 3.2 is of two types. By applying the descent argument of Kolyvagin, one can prove part 1 of the conjecture (for D < 0) under certain mild extra hypotheses. This result will be presented in section 3.4.

The proof of the conjecture for D > 0 would seem to require new ideas, but it is more amenable to numerical verification on the computer, since the module M_{D} in this case is simpler. A summary of some of the computer calculations that were performed is given in section 3.5.

3.2 Properties of θ_{D}

Let ν_{p} denote the map M_{Dp^{2}} ⊗ Z[G_{Dp^{2}}] ⟶ M_{Dp^{2}} ⊗ Z[G_{D}] induced by the natural homomorphism μ_{p} : G_{Dp^{2}} ⟶ G_{D}.

Proposition 3.3 .

1. ν_{p}(θ_{Dp^{2}}) = (a_{p} − Frob_{p} − Frob_{p}^{−1})θ_{D} if (\frac{p}{D_{0}} ) = 1.

2. ν_{p}(θ_{Dp^{2}}) = a_{p}θ_{D} if (\frac{p}{D_{0}} ) = −1,

Proof: This is a direct consequence of proposition 1.12.

Let θ ↦ θ^{∗} denote the involution which sends σ ∈ G_{D} to σ^{−1}, extended by linearity to the group ring Z[G_{D}]. The following result can be viewed as the analogue of the functional equation for the element θ_{D}.

Proposition 3.4 Frob_{N}^{−1}θ_{D}^{∗} = −ϵθ_{D}.

Proof: By proposition 1.10, w_{N}θ_{D} = −ϵθ_{D}, and by proposition 1.11, w_{N}θ_{D} = Frob_{N}^{−1}θ_{D}^{∗}. The result follows.

Let Z be the ring Z[\frac{1}{2}], and let \underline{θ}_{D} denote the image of θ_{D} in M_{D} ⊗Z[G_{D}]. Define the order of vanishing of \underline{θ}_{D} to be the greatest t such that \underline{θ}_{D} belongs to M_{D} ⊗ \underline{I}^{t}.

Proposition 3.5 If ϵ = 1, then \underline{θ}_{D} has odd order of vanishing, and if ϵ = −1, then \underline{θ}_{D} has even order of vanishing.

Proof: Let t denote the order of vanishing, and let \underline{θ̃}_{D} and \underline{θ̃}_{D}^{∗} denote the leading coefficients in M_{D} ⊗ (\underline{I}^{t}/\underline{I}^{t+1}). Since θ̃_{D}^{∗} = (−1)^{t}θ̃_{D}, the functional equation of proposition 3.4 implies that

(−1)^{t}\underline{θ̃}_{D} = −ϵ\underline{θ̃}_{D}.

Since the group M_{D} ⊗ (\underline{I}^{t}/\underline{I}^{t+1}) is of odd order, it follows that

(−1)^{t+1} = ϵ,

which proves the proposition.

3.3 The leading coefficient

Assuming the truth of the order of vanishing conjectures 3.1 and 3.2, define the leading coefficient of θ_{D} to be the projection of θ_{D} to the group M_{D} ⊗ (I^{ρ−1}/I^{ρ}) if D < 0, and to M_{D} ⊗ (I^{ρ}/I^{ρ+1}) if D > 0. It is natural to search for an interpretation of the leading coefficient θ̃_{D} in terms of arithmetic data for the curve E over K. Such a conjecural interpretation can only be given at the moment for the following cases:

1. D < 0, |r^{+} − r^{−}| = 1.

2. D > 0, r^{+} = r^{−}.

(These represent exactly the cases for which conjecture 3.2 is no stronger than conjecture 3.1). The conjecture concerning the value of the leading coefficient in case 1 is stated in full generality in [D2]. We concentrate here on the simpler case where D > 0. To simplify the discussion, we will state the conjecture only in the case where f is a prime > 3 and r > 0, and E(K) is torsion-free. (In the general case, one needs to modify the naive element θ_{D} by a kind of regularization process which is explained in [D2] to make the conjecture compatible under norms and take into account the Euler factors at the primes dividing f.)

In [MT], B. Mazur and J. Tate define a height pairing based on an idea of Manin and Zarhin, which takes values in G_{D} = I/I^{2}. This height pairing is not defined on the full Mordell-Weil groups, but only on E(K) × E_{f}(K), where E_{f}(K) denotes the subgroup of E(K) of finite index in E(K) defined by the exact sequence:

0 ⟶ E_{f}(K) ⟶ E(K) ⟶ E/E^{0}(K) ⊕ E(k_{f}).

Here k_{f} denotes the residue field of K at f. Let further J_{D} denote the order of the cokernel of the right-hand map.

Let P_{1}, …, P_{r} (resp. Q_{1}, … Q_{r}) denote integral bases for E(K) (resp. E_{f}(K)) modulo torsion which induce compatible orientations on E(K) ⊗ R. Let R_{D} be the determinant of r ×r matrix (⟨P_{i}, Q_{j}⟩_{D} with entries in I/I^{2}. It belongs naturally to I^{r}/I^{r+1}. The element R_{D} plays the role of the regulator. It is independent of the choice of bases.

Conjecture 3.6

L̃_{D} (= (−1)^{ρ}θ̃_{D}^{2}) = #\underline{III}(E/K) · R_{D} · J_{D} · ω^{+} ⊗ ω^{+}.

What is the correct generalization of this conjecture when ρ is greater than r/2? In that case conjecture 3.2 predicts that the image of L̃_{D} in M_{D}^{⊗2} ⊗ I^{r} is 0. It can also be shown that the Mazur-Tate regulator R_{D} in I^{r}/I^{r+1} vanishes: for the height pairing is trivial when restricted to E(Q) × E_{f}(Q) or E^{−}(K) × E_{f}^{−}(K), and the hypothesis r^{+} ≠ r^{−} implies that one of the isotropic subspaces has dimension > r/2.

However, one feels strongly that the leading coefficient θ̃_{D} in M_{D}⊗I^{ρ}/I^{ρ+1} should have an arithmetic interpretation in all cases. ^{1}

3.4 The case D < 0: theoretical evidence

We now return to the order of vanishing conjecture 3.2, and state a result which gives evidence for it in the case where D < 0. For details and proofs see [D2].

Suppose that E has no complex multiplications, and supppose that D < 0 is such that all primes dividing f are inert in K = Q(\sqrt{D}).

Let Z be a subring of Q such that the following are invertible:

1Note: Since this paper was submitted, such an interpretation has been found. See the forthcoming publication by Massimo Bertolini and the author on “Derived Height Pairings”.

1. All primes p|6.

2. All primes p < ρ.

3. All primes p such that Gal(Q(E_{p^{∞}})/Q) is smaller than the full group of Z_{p}-linear automorphisms of the Tate module T_{p}(E).

4. All p which divide [E : E^{0}].

By a result of Serre [Se], the set of primes satisfying condition 3 is a finite set (in the case where N is squarefree, it consists at most of the primes ≤ 11). Let \underline{θ}_{D} be the image of θ_{D} in the group M_{D} ⊗ Z[G_{D}], and let \underline{I} denote the augmentation ideal in the group ring Z[G_{D}].

Theorem 3.7 \underline{θ}_{D} belongs to the subgroup M_{D} ⊗ (\underline{I}^{ρ−1}/\underline{I}^{ρ}) of M_{D} ⊗ Z[G_{D}].

A key ingredient in the proof of this result is the descent method of Kolyvagin [Ko1], [Ko2].

3.5 The case D > 0: computational evidence

The conjecture for D > 0 is closer to the original cyclotomic conjecture of Mazur and Tate formulated in [MT] in terms of modular symbols, which remains unproved. One must thus content oneself with numerical verification on the computer. Fortunately, the module M_{D} is much simpler than when D < 0, making such calculations feasible.

Because of the close analogy between Heegner cycles and Heegner points, it is hoped that the numerical study of Heegner cycles will provide insights into the behaviour of Heegner points in the anticyclotomic tower. An understanding of this behaviour is crucial if one wants to extend Kolyvagin’s methods to modular elliptic curves of higher rank.

We now give three representative examples, involving modular elliptic curves of conductor 11, 37 and 5077.

3.5.1 The curve X_{0}(11)

Let E be the modular curve X_{0}(11), given by the equation

y^{2} + y = x^{3} − x^{2} − 10x − 20.

Its Mordell-Weil group has rank 0 over Q, and the eigenvalue of the Atkin-Lehner involution w_{11} is ϵ = −1.

A list of values of D_{0} ≤ 500 which satisfy the Heegner hypothesis for E and for which L(E/K, s) has a (non-trivial) zero at s = 1 can be found in the tables compiled by Glenn Stevens [St], p. 171. They are: D_{0} = 232, 265, 273, 364, 401, 421, 476, 488. If f is prime to 11D_{0}, and if D = D_{0}f^{2}, conjecture 3.2 predicts that

θ_{D} ∈ M_{D} ⊗ I^{2}.

In this case much of it can be proved. Let \underline{θ_{D}} denote as in section 3.2 the image of θ_{D} in M_{D} ⊗ Z[G_{D}], where Z is the ring Z[\frac{1}{2}].

Proposition 3.8 \underline{θ}_{D} belongs to M_{D} ⊗ \underline{I}^{2}.

Proof: Waldspurger’s formula (theorem 2.3) implies that \underline{θ}_{D} belongs to M_{D} ⊗ \underline{I}. The result follows from proposition 3.5 and the fact that ϵ = −1 for the curve X_{0}(11).

A verification of the full vanishing conjecture by computation remains interesting nonetheless, since it enables us to compute the values of the mysterious leading terms θ̃_{D} ∈ M_{D} ⊗ (I^{2}/I^{3}).

A summary of some of these computations is presented in table 1.

1. Column 1 of the table indicates the value of D, written in the form D_{0}f^{2}. In some cases the full θ_{D} was not computed, but only its projection in Z[G_{D}^{(2)}], where G_{D}^{(2)} is the quotient of G_{D} by its 2-primary part. These cases are indicated by an asterisk next to the value of D.

2. Column 2 indicates the order, and structure, of the class group G_{D} by giving the factorization h^{+}(D) = h_{1} ⋯ h_{k}, where

G_{D} = Z/h_{1} × ⋯ × Z/h_{k}Z,

and h_{i} is divisible by h_{j} whenever j > i. The integers h_{i} are determined uniquely by G_{D} and determine the structure of G_{D} completely. In the cases where the 2-part was ignored, the similar factorization is given, but only for the odd part of the class group.

3. Column 3 indicated generators σ_{1}, …, σ_{k} for each of the cyclic pieces of the class group.

4. Column 4 gives the leading coefficient of the θ-element. The expressions σ_{1}, …, σ_{k} refer to the corresponding generators given in column 3. We have omitted the factor of ω^{+} which appears in all of the leading coefficients.

3.5.2 The curve X_{0}(37)^{+}

Let E be the elliptic curve of conductor 37 given by the equation

y^{2} − y = x^{3} − x.

The curve E is modular; in fact, it is the quotient of the modular curve X_{0}(37) of genus 2 by the Atkin-Lehner involution w_{37}, and hence the map ϕ of conjecture 1.1 is of degre 2.

If D satisfies the Heegner hypothesis for E, then conjecture 3.2 predicts that

θ_{D} belongs to M_{D} ⊗ I.

In this case the conjecture follows from the formula (theorem 2.3) of Waldspurger. It is interesting to check that this predicted order of vanishing is sharp, i.e., that the leading coefficients of θ̃_{D} are in general non-trivial in M_{D} ⊗ (I/I^{2}).

Since ρ = r^{+} = r^{−} = 1, conjecture 3.6 of section 3.3 gives a precise prediction for the value of the leading coefficient. We plan to verify this conjecture by finding rational points on the relevant twists of X_{0}(37)^{+} and computing the Mazur-Tate height pairing between them, but have not carried out this computation at present.

The first 10 positive fundamental discriminants which satisfy the Heegner hypothesis are 12, 21, 28, 33, 40, 41, 44, 53, 65, and 73. We have limited our computations to discriminants of the form D_{0}f^{2} where D_{0} is one of these ten fundamental discriminants, and f is prime. The results of the computation are summarized in table 2, with the same conventions as for the previous table.

Table 1: The curve X_{0}(11) D = D_{0}f^{2} h^{+}(D) Generators of G_{D}

Leading term

232 · 13^{2} 14 · 2 (33, 134, −161), 11 · (σ_{1} − 1)^{2}
(69, 142, −69)  +(σ_{1} − 1)(σ_{2} − 1)
232 · 35^{2} 6 · 2 (−3, 530, 275) 4(σ_{1} − 1)^{2}
(−2, 532, 147)
232 · 323^{2} 18 · 2 · 2 (−3, 4916, 3106) 8(σ_{1} − 1)^{2}
(1673, 3116, −2166)
(−1847, 2974, 2079)
232 · 7841^{2} 2614 · 2 (40293, 41732, −77694), 486(σ_{1} − 1)^{2}
(−50919, 84196, 35226)  +(σ_{1} − 1)(σ_{2} − 1)
232 · 146329^{2}(^{∗}) 21 · 21 (9, 2228804, −1600486) 9(σ_{1} − 1)^{2}
(57121, 2174902, −1039137) +15(σ_{2} − 1)^{2}
265 · 373^{2} 372 · 2 (−60, 6065, 354) 294(σ_{1} − 1)^{2}
(−1320, 4465, 3207)
273 · 727^{2} 364 · 2 · 2 (2, 12009, −9042) 284(σ_{1} − 1)^{2}
(−21, 11991, 6004)
(−6028, 12009, 3)
401 · 1601^{2}  2670  (−1502, 29659, 24665)  2520(σ_{1} − 1)^{2}
421 · 97^{2}  96  (15, 1967, −1535)  42(σ_{1} − 1)^{2}
421 · 139^{2}  138  (15, 2849, −289)  2(σ_{1} − 1)^{2}
421 · 331^{2}  166  (15, 6791, −125)  138(σ_{1} − 1)^{2}
421 · 4337^{2}  4338  (−5, 88983, 42713)  3147(σ_{1} − 1)^{2}
488 · 97^{2}  32 · 2  (−118, 2040, 911)  5(σ_{1} − 1)^{2}
(−61, 2074, 1189)  +(σ_{1} − 1)(σ_{2} − 1)

Table 2: The curve X_{0}(37)^{+}

D = D_{0}f^{2} h^{+}(D) Generators of G_{D}

Leading term

12 · 607^{2} 38 · 2 (−1034, 2066, 37)  8 · (σ_{1} − 1)
(3, 2100, −949)
12 · 2131^{2} 164 · 2 (−3351, 6042, 1342)  128(σ_{1} − 1)
(2, 7382, −1)
12 · 3691^{2} 284 · 2 (3489, 8136, −6971)  12(σ_{1} − 1)^{3}
(3, 12780, −12781)
21 · 2089^{2} 190 · 2 (−3617, 4923, 4659)  94(σ_{1} − 1)
(3, 9573, −1)
21 · 3191^{2} 290 · 2 (−1433, 14047, 2881)  34(σ_{1} − 1)
(3, 14619, −9745)
28 · 271^{2} 54 · 2 (498, 538, −887)  36(σ_{1} − 1)
(7, 1428, −613)
28 · 617^{2} 88 · 2 (−3, 3260, 2641)  8(σ_{1} − 1)
(7, 3262, −666)
33 · 151^{2} 38 · 2 (−2, 867, 93)  26(σ_{1} − 1)
(62, 867, −3)
33 · 2069^{2} 414 · 2 (−3284, 7875, 6033)  140(σ_{1} − 1)
(−3, 11883, 4952)
40 · 281^{2} 56 · 2 (−6, 1772, 769)  24(σ_{1} − 1)
(−5, 1770, 1277)
40 · 3221^{2} 358 · 2 (−7734, 10576, 9799)  102(σ_{1} − 1)
(−5, 20370, 2837)
41 · 241^{2} 80  (−206, 1347, 688)  128(σ_{1} − 1)
41 · 1493^{2} 166  (2, 9557, −6845)  128(σ_{1} − 1)
41 · 99241^{2} 6616 (−1112, 634453, 285344) 5808(σ_{1} − 1)
44 · 199^{2} 50 · 2 (−10, 1318, 133)  18(σ_{1} − 1)
(11, 1320, −1)
44 · 379^{2} 76 · 2 (−659, 1460, 1589)  68(σ_{1} − 1)
(11, 2508, −685)
44 · 419^{2} 84 · 2 (−7, 2768, 2245)  60(σ_{1} − 1)
(11, 2772, −925)

Table 2 (cont’d): The curve X_{0}(37)^{+}.

D = D_{0}f^{2} h^{+}(D) Generators of G_{D}

Leading term

53 · 2549^{2} 510  (−11, 18537, 16861) 40(σ_{1} − 1)
53 · 3581^{2} 398  (−11, 26055, 17857) 186(σ_{1} − 1)
65 · 89^{2} 18 · 2 (244, 249, −464) 4(σ_{1} − 1)
(−5, 715, 182)
65 · 149^{2} 30 · 2 (−10, 1185, 971)  14(σ_{1} − 1)
(−5, 1195, 752)
65 · 181^{2} 20 · 2 (−10, 1445, 1036) 8(σ_{1} − 1)
(−5, 1455, 622)
65 · 257^{2} 86 · 2 (−14, 2059, 959)  38(σ_{1} − 1)
(−5, 2065, 1448)
65 · 353^{2} 32 · 2 (−14, 2823, 2326) 8(σ_{1} − 1)
(−5, 2845, 278)
65 · 449^{2} 30 · 2 (−1316, 1663, 1964) 12(σ_{1} − 1)
(−5, 3615, 1792)
73 · 1901^{2} 380  (−12, 16225, 11601) 232(σ_{1} − 1)

3.5.3 The curve of conductor 5077

Let E be the elliptic curve of conductor 5077 given by the equation

y^{2} + y = x^{3} − 7x + 6.

It was proved by Mestre [Me], p. 232, that E is modular. (In fact, the degree of the map ϕ of conjecture 1.1 was computed by Zagier [Z], and is equal to 1984.)

If D satisfies the Heegner hypothesis for E, then conjecture 3.2 predicts that

θ_{D} belongs to M_{D} ⊗ I^{3}.

This has been checked for a number of values of D; the results of the computation are summarized in table 3.

Table 3: The curve of conductor 5077

D = D_{0}f^{2} h^{+}(D) Generators of G_{D}

Leading term

12 · 71^{2} 10 · 2 (13, 238, −74) 0
(2, 242, −241)
21 · 2089^{2} 190 · 2 (−1915, 5749, 7649) 122(σ_{1} − 1)^{3}
(7, 9669, −2735)
21 · 3191^{2} 290 · 2 (3527, 13205, −2797) 286(σ_{1} − 1)^{3}
(−1, 14621, 14615)
28 · 17^{2} 6 · 2 (3, 86, −58) 0
(−27, 44, 57)
53 · 2549^{2} 510  (6773, 12507, −6937) 194(σ_{1} − 1)^{3}
53 · 4481^{2} 640  (97, 32603, −3223) 560(σ_{1} − 1)^{3}
53 · 3581^{2} 398  (211, 25745, −19957) 393(σ_{1} − 1)^{3}
57 · 151^{2} 76 · 2 (2, 1137, −861)  4(σ_{1} − 1)^{3}
(−1, 1139, 584)
61 · 761^{2} 254 (3, 5941, −2575) 30(σ_{1} − 1)^{3}
61 · 5209^{2} 1042 (827, 39703, −23829) 472(σ_{1} − 1)^{3}
61 · 3373^{2} 482  (19, 26335, −6219) 198(σ_{1} − 1)^{3}
65 · 257^{2} 86 · 2 (2, 2069, −1553)  8(σ_{1} − 1)^{3}
(−622, 1657, 622)
76 · 113^{2} 38 · 2 (3, 980, −837)  10(σ_{1} − 1)^{3}
(2, 982, −765)
85 · 1361^{2} 272 · 2 (3, 12547, −1673)  32(σ_{1} − 1)^{3}
(5, 12545, −3513)
88 · 197^{2} 98 · 2 (3, 1844, −1238)  8(σ_{1} − 1)^{3}
(22, 1848, −1)
89 · 53^{2}  52  (2, 497, −374)  8(σ_{1} − 1)^{3}
89 · 101^{2} 34  (2, 949, −911)  4(σ_{1} − 1)^{3}
97 · 569^{2} 570  (388, 5529, −538)  482(σ_{1} − 1)^{3}
97 · 401^{2} 134  (3, 3949, −258)  4(σ_{1} − 1)^{5}

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