March 29, 2011 10:4 WSPC - Proceedings Trim Size: 9in x 6in oda

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The Birch and Swinnerton-Dyer conjecture for ℚ-curves and Oda’s period relations

Henri Darmon

Department of Mathematics and Statistics, McGill University, 805 Sherbooke Street West, Montreal, Canada H3A 2K6 E-mail: darmon@math.mcgill.ca

Victor Rotger ^{∗}

Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, C.

Jordi Girona 1-3, 08034 Barcelona, Spain E-mail: victor.rotger@upc.edu

Yu Zhao

Department of Mathematics and Statistics, McGill University, 805 Sherbooke Street

West, Montreal, Canada H3A 2K6 E-mail: zhao@math.mcgill.ca

Let N ≡ 1(mod 4) be a square-free positive integer, let ε be the primitive quadratic character of conductor N, and let f ∈ S_{2}(Γ_{0}(N), ε) be a newform with fourier coefficients in a quadratic field. Shimura associates to f an elliptic curve E defined over the real quadratic field F = ℚ(\sqrt{N}) which is isogenous over F to its Galois conjugate. Let M/F be a quadratic extension of F which is neither CM or totally real, and denote by E_{M} the twist of E with respect to M/F. The first main result of this article is that, if L′(E_{M} /F, 1) ≠ 0, then E_{M} (F) has rank one and LLI(E/F) is finite. The proof rests on a fundamental result^{42} of Yuan, Zhang and Zhang and on an explicit Heegner point in E_{M} (F) arising from the modular parametrization X_{1}(N) → E. We formulate a conjecture relating this Heegner point to Stark-Heegner points^{8} arising from ATR cycles of real dimension one on Hilbert modular surfaces, and present some numerical evidence for this conjecture.

Keywords: ℚ-curves; Birch and Swinnerton-Dyer conjecture; Oda’s period relations; Heegner points; Stark-Heegner points; ATR cycles; Hilbert modular forms

∗The research of the second author is financially supported by DGICYT Grant MTM2009-13060-C02-01 and the Grup de recerca consolidat de Catalunya 2009 SGR 2

1. Introduction

Let E be an elliptic curve over a number field F and let L(E/F, s) denote its Hasse-Weil L-series. It is widely believed that the Shafarevich-Tate group LLI(E/F) is finite and that L(E/F, s) extends to an entire function of the complex variable s. The order of vanishing of this function at s = 1, denoted by r_{an}(E/F), is commonly referred to as the analytic rank of E over F, a terminology justified by the Birch and Swinnerton-Dyer conjecture which asserts that

rank(E(F)) =^{?} r_{an}(E/F).  (1)

The most convincing evidence for the Birch and Swinnerton-Dyer conjecture is the fact that it is proved when F = ℚ and L(E, s) := L(E/ℚ, s) has at most a simple zero at s = 1:

Theorem 1.1 (Gross-Zagier, Kolyvagin). If r_{an}(E/ℚ) ≤ 1, then (1) holds for E/ℚ, and LLI(E/ℚ) is finite.

The proof of Theorem 1.1, which is briefly recalled in Section 2.1, rests on two key ingredients. The first is the modularity of E, in the strong geometric form which asserts that E is a quotient of the Jacobian of a modular curve over ℚ. The second is the collection of Heegner points on this modular curve, which satisfies the axioms of an “Euler system” and provides a valuable bridge between the arithmetic of E and the analytic behaviour of its L-series.

Both these ingredients are available in greater generality, most notably when F is a totally real field. In this setting, a modular elliptic curve E over F is said to satisfy the Jacquet-Langlands hypothesis (JL) if either [F : ℚ] is odd, or there is at least one prime of F at which the automorphic form on GL_{2}(𝔸_{F}) attached to E is not in the principal series. Here, 𝔸_{F} stands for the ring of adèles of F. The meaning of condition (JL), which only fail to hold for certain elliptic curves of square conductor, is described more concretely in Section 2.1.

Most importantly for the proof of (1), the Jacquet-Langlands hypothesis implies that E is the quotient of the Jacobian of a suitable Shimura curve over F. Shimura curves are equipped with a plentiful supply of CM points, which have been parlayed into the proof of the following number field generalisation of Theorem 1.1.

Theorem 1.2 (Zhang). Let E be a modular elliptic curve over a totally real field F satisfying (JL). If r_{an}(E/F) ≤ 1, then LLI(E/F) is finite and (1) holds for E/F.

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Denote by 𝔑 the conductor of E/F. If either [F : ℚ] is odd or there exists a prime 𝔭 of F for which ord_{𝔭}(𝔑) = 1, this is just [43, Theorem A]. The full result in which E is only assumed to satisfy (JL) follows from the subsequent strengthening of the Gross-Zagier formula^{42} of Yuan, Zhang and Zhang. Both results are discussed further in Section 2.1.

In analytic rank zero, the Jacquet-Langlands hypothesis can be dispensed with:

Theorem 1.3 (Longo). Let E be a modular elliptic curve over a totally real field F. If L(E/F, 1) ≠ 0, then E(F) and LLI(E/F) are finite.

The proof^{24} of Longo, extending to totally real fields an earlier treatment^{2} of the case F = ℚ, exploits the theory of congruences between modular forms to realise the Galois representation E[p^{n}] in the Jacobian of a Shimura curve X_{n} whose level may (and indeed does) depend on n. The Euler system of CM points on X_{n} then gives rise to a collection of p^{n}- torsion cohomology classes which is used to bound the p^{n}-Selmer group of E over F independently of n, and thereby obtain the finiteness of E(F) and LLI(E/F).

The problem of removing the Jacquet-Langlands hypothesis from Theorem 1.2—or equivalently, of extending Theorem 1.3 to the case where L(E/F, s) has a simple zero at s = 1—is still very much open.

To better understand the difficulty which arises, it is instructive to examine the simplest setting where the Jacquet-Langlands hypothesis fails to hold. Assume for the rest of the introduction that F is a real quadratic field, and consider for now the case where E/F is an elliptic curve of conductor 1.

Assuming E is modular, the L-series L(E/F, s) is known to have a functional equation relating its values at s and 2 − s, and the sign w_{E} ∈ {−1, 1} in this functional equation is always equal to 1 in this case.

Let M be any quadratic extension of F, let

χ_{M} : G_{F} ⟶ ±1

be its associated Galois character, and denote by E_{M} the twist of E over F by χ_{M}, so that the L-series L(E/M, s) factors as

L(E/M, s) = L(E/F, s)L(E/F, χ_{M}, s) = L(E/F, s)L(E_{M}/F, s).

Since E has conductor 1, the sign w_{E_{M}} of the twisted L-series is controlled by the local signs attached to the archimedean places ∞_{1} and ∞_{2} of F, which are equal to χ_{M}(∞_{1}) and χ_{M}(∞_{2}) respectively. It follows that

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w_{E_{M}} = 1 if M is either totally real or CM. In particular, the elliptic curve E is always of even analytic rank over such M. Since an Euler system of Heegner points attached to a quadratic CM extension M/F is only expected to be available when E has odd analytic rank over M, this suggests that the mathematical objects so crucial in Kolyvagin’s descent method may be unavailable for elliptic curves of conductor 1.

A similar expectation can be derived more generally for all elliptic curves which do not satisfy (JL). Indeed, if E/F is an elliptic curve of square conductor 𝔑 and M is a quadratic extension of F which is unramified at the primes dividing 𝔑, then the same analysis as above reveals that

r_{an}(E/M) ≡ \{ \begin{matrix}0 (mod 2), & if M is CM or totally real; \\ 1 (mod 2) & otherwise. \end{matrix}  (2)

A quadratic extension M of F which is neither CM nor totally real is called an ATR extension of F. An ATR extension of F thus has two real places and one complex place. (The acronym “ATR” stands for “Almost Totally Real” and is used more generally^{8} to designate quadratic extensions of a totally real field having exactly one complex place.)

The present article is motivated by the following specific instance of the Birch and Swinnerton-Dyer conjecture which emerges naturally from the discussion above.

Conjecture 1.1. Let E be a (modular) elliptic curve over a real quadratic field F of square conductor 𝔑 for which w_{E} = 1, and let M/F be an ATR extension of F of discriminant prime to 𝔑. If L′(E_{M}/F, 1) ≠ 0, then E_{M}(F) has rank one and LLI(E_{M}/F) is finite.

Although it seems tantalisingly close to the setting of Theorem 1.2, Conjecture 1.1 presents a real mystery and appears to lie beyond the reach of known methods. The difficulty is that, in the absence of the Jacquet-Langlands hypothesis, no natural “modular” method presents itself in general for constructing the point of infinite order on E_{M}(F) whose existence is predicted by the Birch and Swinnerton–Dyer conjecture.

One of the original motivations for singling out Conjecture 1.1 for special study lies in the conjectural construction^{8} of a so-called Stark-Heegner point P_{M}^{?} ∈ E(M). This construction, which is recalled briefly in Section 2.2, involves the images under a complex Abel-Jacobi map attached to the Hilbert modular form associated to E/F of certain “ATR cycles” indexed by ideals of M. The ATR cycles are null-homologous cycles of real dimension one on the corresponding Hilbert modular surface. It is conjectured^{8}

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that the point P_{M}^{?} is of infinite order precisely when L′(E/M, 1) ≠ 0, and that P_{M}^{?} is part of a norm-coherent collection of points defined over abelian extensions of M satisfying Euler-System-like properties. However, progress on Conjecture 1.1 through the theory of ATR cycles is thwarted by our inability to provide much theoretical evidence for the algebraicity of ATR points.

The first aim of this note is to study Conjecture 1.1 for the class of elliptic curves E/F which are isogenous over F to their Galois conjugate. Following a terminology that was first introduced^{32} by Ribet, these elliptic curves are called ℚ-curves. Their basic properties are reviewed in Section 3. As explained in that section, the case of ℚ-curves is ultimately made tractable by the existence of a classical elliptic cusp form f (with nontrivial nebentypus character in general) satisfying

L(E/F, s) = L(f, s)L(f̄, s),

leading to a modular parametrisation of E by a classical modular curve X_{1}(N) for a suitable N ≥ 1. The main theorem of Section 3 is a proof of Conjecture 1.1 for ℚ-curves:

Theorem 1.4. Let E/F be a ℚ-curve of square conductor 𝔑, and let M/F be an ATR extension of F of discriminant prime to 𝔑. If L′(E_{M}/F, 1) ≠ 0, then E_{M}(F) has rank one and LLI(E_{M}/F) is finite.

The key ingredients in the proof of Theorem 1.4 are a strikingly general recent extension of the theorem of Gross-Zagier obtained by Xinyi Yuan, Shouwu Zhang and Wei Zhang^{42} covering cusp forms with possibly nontrivial nebentypus characters, and a strengthening of Kolyvagin’s descent method to cover abelian variety quotients of J_{1}(N), as worked out in the forthcoming book^{39} of Ye Tian and Shouwu Zhang. Section 3 explains how Theorem 1.4 follows from these results and the Artin formalism for certain Rankin L-series.

The second part of the article focuses on the special case where the ℚ-curve E is of conductor 1. Such elliptic curves, which were first systematically studied^{35} by Shimura, are essentially in bijection with newforms f in S_{2}(Γ_{0}(N), ε_{N}) with quadratic Fourier coefficients, where N is the discriminant of the real quadratic field F, and

ε_{N} : (ℤ/Nℤ)^{×} ⟶ ±1

is the corresponding even Dirichlet character.

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Section 4 describes the explicit construction, for all quadratic ATR extensions M of F, of a canonical point P_{M} ∈ E(M) arising from suitable CM divisors on X_{1}(N). The trace to E(F) of P_{M} is shown to vanish, so that P_{M} can also be viewed as an F -rational point on the twisted curve E_{M}.

After explaining how the points P_{M} can be computed complex analytically by integrating the elliptic modular form f, we tabulate these points for a few ATR extensions M of small discriminant. One expects that the height of the point P_{M} is related in a simple way to L′(E_{M}/F, 1).

Finally, Conjecture 4.1 spells out a precise conjectural relationship between the classical Heegner point P_{M} and the Stark-Heegner point P_{M}^{?} arising from ATR cycles on the Hilbert modular variety. This conjecture, which relates certain complex analytic invariants attached to an elliptic modular form f and its Doi-Naganuma lift, can be viewed as an analogue for Abel-Jacobi maps of Oda’s period relations which are studied^{28} in Oda’s influential monograph. It is therefore a pleasure to dedicate this article to Takayuki Oda whose work on periods of Hilbert modular surfaces was a major source of inspiration for the conjectures^{8} on ATR points.

It is also a pleasure to thank Xavier Guitart, Ariel Pacetti and David Rohrlich for their comments on a previous version of this manuscript.

2. Background

2.1. The Birch and Swinnerton-Dyer conjecture in low analytic rank

We begin by recalling in greater detail the main ideas behind the proofs of Theorems 1.1 and 1.2. We start with the assumption that F is a number field and write O_{F} for its ring of integers. Let 𝔑 ⊂ O_{F} denote the conductor of the elliptic curve E/F.

The proofs of Theorems 1.1 and 1.2 can be broken up into five steps:

(i) Modularity: When F = ℚ, the main results of Wiles^{41} and Tatlor-Wiles,^{38} as completed^{1} by Breuil, Conrad, Diamond and Taylor, imply that there is a normalised newform f of weight 2 on Γ_{0}(N) satisfying L(E, s) = L(f, s). In particular, L(E, s) has an analytic continuation to the left of the half plane Re(s) > 3/2, and its order of vanishing at s = 1 is therefore well defined. For general F, the modularity of E/F is just the assertion that L(E/F, s) is the L-series attached to an automorphic representation of GL_{2}(𝔸_{F}). Such a property is predicted to hold, as a (very special) case of the Langlands functoriality conjectures. In spite of the pow-

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erful ideas introduced into the subject building on Wiles’ breakthrough, a proof in the general number field setting still seems a long way off. When F is totally real, modularity can be phrased in terms of modular forms much as in the case F = ℚ. Namely, E/F is modular whenever there is a normalised Hilbert modular eigenform f of parallel weight 2 on the congruence group Γ_{0}(𝔑) ⊆ SL_{2}(O_{F}) satisfying L(E/F, s) = L(f, s). The methods originating from Wiles’ work seem well suited to yield a proof of modularity of all elliptic curves over totally real fields. (See for example the works of Skinner-Wiles,^{37} Fujiwara,^{10} Jarvis-Manoharmayum^{19} and the references therein for an overview of the significant progress that has been achieved in this direction.) Currently, the case which offers most difficulties arises when the residual Galois representation at 3 is reducible.

(ii) Geometric modularity: Thanks to the geometric construction of Eichler-Shimura and to Faltings’ proof of the Tate conjecture for abelian varieties over number fields, the modularity of E in the case where F = ℚ can be recast as the statement that E is a quotient of the jacobian J_{0}(N) of the modular curve X_{0}(N) over ℚ, where 𝔑 = (N), N ≥ 1. A non-constant morphism

π_{E} : J_{0}(N) ⟶ E  (3)

of abelian varieties over ℚ is called a modular parametrisation attached to E.

When F is a totally real field and E/F is known to be modular, the modular parametrisation arising from Eichler-Shimura theory admits no counterpart in general. However, such a modular parametrisation can be obtained when the Jacquet-Langlands hypothesis formulated in the introduction holds. More precisely, as it is explained in [43, §3], hypothesis (JL) implies that E is a quotient of the Jacobian of a suitable Shimura curve X attached to an order in a quaternion algebra over F which splits at exactly one archimedean place of F. That is, there is a non-constant map

π_{E} : J(X) ⟶ E  (4)

of abelian varieties over F generalising (3). The condition that the automorphic form π = ⊗π_{v} attached to E be a principal series representation at a place v of F is satisfied precisely when E acquires good reduction over an abelian extension of F_{v}. For v ∤ 2, the meanings of various conditions on the local representations π_{v} in terms of the behaviour of E over F_{v} are summarised in the table below.

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π_{v}  E/F_{v}  ord_{v}(𝔑)

Unramified principal series

Good reduction over F_{v} 0

Principal series Good reduction over an abelian extension of F_{v}

even

Steinberg Potentially multiplicative reduction over F_{v}

1 or 2

Supercuspidal Otherwise

≥ 2

(5)

We refer the reader to Gelbart [12, p. 73], Pacetti,^{30} and Rohrich [33, Prop. 2], [34, Prop. 2 and 3] for proofs of these statements. (Note that, although in the latter article the ground field is assumed to be F = ℚ, the results remain valid for arbitrary F as the questions at issue are purely local). See Pacetti^{30} for the behaviour at places v above 2.

In particular, an elliptic curve which fails to satisfy hypothesis (JL) is necessarily of square conductor. The converse is not true, but note that it also follows from the table that all elliptic curves of conductor 1 over a totally real number field of even degree fail to satisfy (JL). We will often restrict our attention to elliptic curves of square conductor, thus encompassing all elliptic curves that do not satisfy hypothesis (JL).

(iii) Heegner points and L-series: Suppose first that F = ℚ and let K be an imaginary quadratic field of discriminant relatively prime to N and satisfying the

Heegner hypothesis: O_{K} has an ideal N of norm N satisfying O_{K}/N ≃ ℤ/Nℤ.

An ideal N of K with this property is sometimes called a cyclic ideal of norm N. When the Heegner hypothesis is satisfied, it can be shown that the L-function L(E/K, s) has sign −1 in its functional equation, and therefore vanishes at s = 1. The CM points on X_{0}(N) attached to the moduli of elliptic curves with complex multiplication by O_{K}, and their images under π_{E}, can be used to construct a canonical point P_{K} ∈ E(K): the so-called Heegner point on E attached to K. The main result of Gross-Zagier^{15} expresses L′(E/K, 1) as a multiple by a simple non-zero scalar of the Néron-Tate height of P_{K}. In particular, the point P_{K} is of infinite order if and only if L′(E/K, 1) ≠ 0.

In the setting where F is a totally real field, the Shimura curve X is

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equipped with an infinite supply of CM points enjoying properties similar to their counterparts on modular curves. The auxiliary field K is now a totally complex quadratic extension of F satisfying a suitable Heegner hypothesis relative to X. The CM points attached to K can be used to construct a canonical point P_{K} ∈ E(K) as in the case F = ℚ. A general extension of the Gross-Zagier theorem [43, Theorem C] to this context relates the height of P_{K} to the derivative L′(E/K, 1). In particular, the point P_{K} is of infinite order precisely when L(E/K, s) has a simple zero at s = 1. We emphasise that this more general Heegner point construction relies crucially on E/F satisfying hypothesis (JL).

(iv) The Euler system argument: The Heegner point P_{K} does not come alone, but can be related to the norms of algebraic points on E defined over abelian extensions of K. Using this fact, it is shown^{20} in the case F = ℚ that the point P_{K}, when it is of infinite order, necessarily generates E(K)⊗ℚ. Koyvagin’s argument extends without essential difficulties to the context of Shimura curves over totally real fields (cf. Kolyvagin-Logachev,^{22} Zhang [43, §7.2], or the forthcoming book^{39} of Tian-Zhang).

(v) Descending from K to F: Assume first that F = ℚ. If ord_{s=1}(L(E, s)) ≤ 1, the analytic non-vanishing results of Bump-Friedberg-Hoffstein^{3} or Murty-Murty^{27} produce an imaginary quadratic field K satisfying the Heegner hypothesis, and for which L(E/K, s) has a simple zero at s = 1. By the Gross-Zagier theorem, the Heegner point P_{K} generates E(K), and its trace therefore generates E(ℚ). The known properties of the Heegner point P_{K} imply in particular that its trace to ℚ vanishes when L(E, 1) ≠ 0, and is of infinite order when L(E, 1) = 0. Theorem 1.1 for E/ℚ follows from this. The proof of Theorem 1.2 is deduced similarly, by noting that the analytic non-vanishing results of Bump, Friedberg and Hoffstein^{3} or Waldspurger^{40} generalize to any number field and again produce a totally complex imaginary quadratic extension K/F satisfying the Heegner hypothesis for which ord_{s=1}(L(E/K, s)) = 1.

2.2. Oda’s period relations and ATR points

. This section briefly recalls the main construction^{8} which to any ATR extension M of F (satisfying a suitable Heegner condition) associates a point P_{M} ∈ E(ℂ) belonging conjecturally to E(M). The points P_{M} arise by considering the images of certain non-algebraic cycles on Hilbert modular varieties under a map which is formally analogous to the Grffiths-Weil Abel-Jacobi maps on null-homologous algebraic cycles.

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The general setting. We begin by treating a more general context where F is a totally real field of degree r+1. (This extra generality does not unduly complicate the notations, and may even clarify some of the key features of the construction.) 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).

(For a more general treatment where these assumptions are significantly relaxed, see for instance the forthcoming PhD thesis^{11} of Jérôme Gärtner.)

The Hilbert modular form G on SL_{2}(O_{F}) attached to E 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 transformation property, the Hilbert modular form G 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}),  (6)

by setting

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

It is important to replace ω_{G}^{hol} by a closed, but non-holomorphic differential (r+1)-form ω_{G} on X(ℂ). When r = 1, the differential ω_{G} 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_{G} = (2πi)^{2} (G(τ_{0}, τ_{1})dτ_{0}dτ_{1} − G(ϵ_{0}τ_{0}, ϵ_{1}τ̄_{1})dτ_{0}dτ̄_{1}) .

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For general r, one defines ω_{G} 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_{G} is holomorphic in τ_{0}, but only harmonic in the remaining variables τ_{1}, …, τ_{r}. The justification for working with ω_{G} rather than ω_{G}^{hol} lies in the following statement which is a reformulation of a conjecture of Oda^{28} in the special case of modular forms with rational fourier coefficients:

Conjecture 2.1 (Oda). Let

Λ_{G} := {∫_{γ} ω_{G}, γ ∈ H_{r+1}(X(ℂ), ℤ)} .

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

This conjecture is shown to hold^{28} for Hilbert modular forms which are base change lifts of classical elliptic modular forms, which corresponds to the case where the associated elliptic curve E is a ℚ-curve. But it should be emphasised that no ℚ-curve hypothesis on E is necessary in Conjecture 2.1.

Let

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

Conjecture 2.1 makes it possible to define an “Abel-Jacobi map”

AJ_{G} : Z_{r}(X(ℂ)) ⟶ E_{0}(ℂ),  (7)

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

AJ_{G}(Δ) := ι (∫Δ̃ ω_{G}) ,  (for any Δ̃ with ∂Δ̃ = Δ).  (8)

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

ATR Cycles. Generalising slightly the definitions given in the Introduction to the case r > 1, a quadratic extension M of F is called an ATR extension if

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

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

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

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

(2) For each 1 ≤ j ≤ r, the fact that M ⊗_{F,v_{j}} ℝ ≃ ℝ⊕ℝ shows that Ψ(M^{×}) 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 : M ⟶ M_{2}(F) has a conductor, which is defined to be the O_{F} -ideal c_{Ψ} for which

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

The O_{F} -order O_{Ψ} := O_{F} +c_{Ψ}O_{M} is called the order associated to Ψ. It can be shown that there are finitely many distinct SL_{2}(O_{F})-conjugacy classes of embeddings of M into M_{2}(F) associated to a fixed order O ⊂ O_{M}, and that the Picard group (in a narrow sense) of O acts simply transitively on the set of such conjugacy classes of embeddings.

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 rdimensional real torus. In many cases, one can show that Δ_{Ψ} is nullhomologous, at least after tensoring with ℚ to avoid the delicate issues arising from the possible presence of torsion in integral homology. (The homological triviality of Δ_{Ψ} always holds, for instance, when r = 1, and follows from the fact that the group cohomology H^{1}(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_{G} and to the ATR cycles Δ_{Ψ}.

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Conjecture 2.2. Let Ψ : M ⟶ M_{2}(F) be an F -algebra embedding of an ATR extension M of F. Then the complex point AJ_{G}(Δ_{Ψ}) ∈ E_{0}(ℂ) is algebraic. More precisely, the isogeny ι in the definition (8) of AJ_{G} can be chosen so that, for all Ψ,

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

where H_{c_{Ψ}} is the ring class field of M of conductor c_{Ψ}. Furthermore, if Ψ_{1}, …, Ψ_{h} is a complete system of representatives for the SL_{2}(O_{F})- conjugacy classes of embeddings of M in M_{2}(O_{F}) of a given conductor c, then the Galois group Gal(H_{c}/M) acts (transitively) on the set {AJ_{G}(Δ_{Ψ_{1}}), …, AJ_{G}(Δ_{ψ_{h}})}.

Conjecture 2.2 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 2.2 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.

Most germane to the concerns of this paper, Conjecture 2.2 can be used as a basis for the construction of a point P_{M}^{?} ∈^{?} E(M), by letting Ψ_{1}, …, Ψ_{h} be a complete system of representatives for the SL_{2}(O_{F})-conjugacy classes of embeddings of M in M_{2}(O_{F}) of conductor 1 and setting

P_{M}^{?} := AJ_{G}(Δ_{Ψ_{1}}) + ⋯ + AJ_{G}(Δ_{Ψ_{h}}).  (9)

3. The Birch and Swinnerton-Dyer conjecture for ℚ-curves

3.1. Review of ℚ-curves

The first goal of the present work is to study Conjecture 2.2 for ℚ-curves, which are defined as follows:

Definition 3.1. Let F be a number field and fix an algebraic closure ℚ̄ of ℚ containing F. We say that an elliptic curve E/F is a ℚ-curve if it is isogenous over F to all its Galois conjugates over ℚ.

In the literature, these curves are sometimes referred as ℚ-curves completely defined over F, reserving the term ℚ-curve for the wider class of elliptic curves over F which are isogenous over ℚ̄ to all their Galois conjugates over ℚ.

14

ℚ-curves are known to be modular, thanks to the work of Ellenberg and Skinner [ES] (who proved (geometric) modularity of ℚ-curves under local conditions at 3), now vastly superseded by Khare-Wintenberger^{23} which implies modularity of all ℚ-curves as a very particular case. Combining this with the older work of Ribet (cf. the survey^{32}), it follows that ℚ-curves E/F are arithmetically uniformisable over ℚ̄ by the classical modular curves X_{1}(N). By this, we mean that there exists a non-constant morphism of curves

π_{E} : X_{1}(N)ℚ̄ ⟶ Eℚ̄  (10)

over ℚ̄, for some N ≥ 1.

For simplicity, in this article we shall restrict our attention to ℚ-curves over a quadratic field F, which represents the simplest non-trivial scenario. However, we believe that the ideas present in this note should allow, with some more effort, to treat more general cases; see the forthcoming Ph.D thesis^{45} of the third author.

Let F be a quadratic field with ring of integers O_{F} and write Gal (F/ℚ) = {1, τ}. Let E be a ℚ-curve over F of conductor 𝔑 ⊂ O_{F}.

Given a Dirichlet character ε of conductor N, let

Γ_{ε}(N) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) | N|c, ε(a) = 1\} ⊆ SL_{2}(ℤ)

and let X_{ε}(N) be the modular curve associated to this congruence subgroup. The curve X_{ε}(N) admits a canonical model over ℚ, and coarsely represents the moduli problem of parametrizing triples (A, C, t) where A is a generalised elliptic curve, C is a cyclic subgroup of order N of A(ℚ̄) and t is an orbit in C \ {O} for the action of ker(ε) ⊂ (ℤ/Nℤ)^{×}. Note that the group (ℤ/Nℤ)^{×} acts on X_{ε}(N) via the diamond operators, and that the subgroup ker(ε) fixes it.

As discussed for example in Guitart-Quer^{14} or Ribet,^{32} the modular parametrisation (10) is particularly well-behaved when F is quadratic. More precisely, there exists a positive integer N ≥ 1, an even Dirichlet character ε : (ℤ/Nℤ)^{×} → {±1} ⊂ ℂ^{×}, and a pair f_{E}, f′_{E} ∈ S_{2}(Γ_{ε}(N)) ⊆ S_{2}(Γ_{0}(N), ε) of normalised newforms of weight 2, level N and nebentypus ε, such that

L(E/F, s) = L(f_{E}, s) · L(f′_{E}, s).  (11)

In this case, the uniformisation in (10) factors through a modular parametrisation

π_{E} : X_{ε}(N)_{F} ⟶ E_{F}  (12)

defined over F.

15

Let K_{f} denote the field generated by the Fourier coefficients of f_{E}. It is either ℚ or a quadratic field. When K_{f} = ℚ, the elliptic curve E is in fact isogenous to the base change of an elliptic curve defined over ℚ and question (1) can rather be tackled with the classical techniques reviewed in §2; we assume throughout that this is not the case. Hence [K_{f} : ℚ] = 2 and, letting σ denote the single nontrivial automorphism of K_{f}, we have

f′_{E} = ^{σ}f_{E}.

Weil’s restriction of scalars A := Res_{F/ℚ}(E) is an abelian surface of GL_{2}-type over ℚ such that

End_{ℚ}(A) ⊗ ℚ ≃ K_{f} (and thus is simple over ℚ),  (13)
A_{/F} ≃ E ×^{τ} E, and  (14)
L(A/ℚ, s) = L(E/F, s) = L(f_{E}, s) · L(f′_{E}, s).  (15)

Moreover, for any field extension L/ℚ, there is a canonical isomorphism

A(L) ≃ E(F ⊗_{ℚ} L)  (16)

and in particular A(ℚ) ≃ E(F). As shown by Carayol, the conductor of A over ℚ is N^{2}, and it follows from [25, Prop. 1] (see also [13, Remark 9] for a more detailed discussion) that the conductor of E/F is 𝔑 = N_{0} · O_{F}, where N_{0} ∈ ℤ satisfies

N = N_{0} · |disc(F)|.  (17)

As we shall now explain, when F is imaginary the problem can be reduced to the classical setting considered by Gross-Zagier and Kolyvagin-Logachev, and presents no mysteries. It is the case of F real that deserves more attention, and to which the main bulk of this note will be devoted.

If ε is trivial, then K_{f} is real and F can be either real or imaginary (and indeed both cases occur in examples). As a direct consequence of (11), (16) and the generalization^{21} of the work of Kolyvagin to higher dimensional quotients of J_{0}(N) over ℚ, (1) also holds for E/F provided ord_{s=1}L(f_{E}, s) ≤ 1.

Assume ε ≠ 1 for the rest of this article. Now K_{f} is an imaginary quadratic field. Besides, it follows from an observation of Serre (cf. [32, Proposition 7.2]) that F is necessarily real. In fact, F can be computed explicitly from f_{E} as F = ℚ̄^{ker(ε)}. In particular it follows that ε is the quadratic Dirichlet character associated with F.

16

Let ω_{N} denote the Fricke involution of X_{ε}(N)_{F} defined on the underlying Riemann surface by the rule τ ↦ −\frac{1}{Nτ}. It induces an involution on the jacobian J_{ε}(N)_{F} of X_{ε}(N)_{F} which leaves A_{F} stable. We have

A_{F} ∼ (1 + ω_{N})A_{F} × (1 − ω_{N})A_{F} ,  (18)

where both factors on the right have dimension 1, are isogenous over F and conjugate one each other over ℚ (cf. [6, §5]). By replacing E by its conjugate if necessary, we shall assume throughout that E = (1 + ω_{N})A_{F}.

It then follows that (12) factors through the following commutative diagram:

\begin{matrix}X_{ε}(N)_{F} & ⟶ & J_{ε}(N)_{F} & ⟶ & A_{F} \\ ↓ π_{N} & & ↓ & & ↓ \\ X_{ε}^{+}(N) & ⟶ & J_{ε}^{+}(N) & ⟶^{φ_{E}} & E, \end{matrix}  (19)

where we set X_{ε}^{+}(N) := X_{ε}(N)_{F} /⟨ω_{N}⟩ and J_{ε}^{+}(N) := (1 + ω_{N})J_{ε}(N)_{F}. The reader should keep in mind that both are varieties over F, not over ℚ.

3.2. The main result

The goal of this section is to prove Theorem 1.4 of the Introduction. Let M be a quadratic ATR extension of F. Since M has two real places and one complex place, it is not Galois over ℚ. Let M′ denote its Galois conjugate over ℚ, and let M be the Galois closure of M over ℚ. It is not hard to see that M is the compositum over F of M and M′ and that Gal(M/ℚ) is isomorphic to the dihedral group of order 8. The subgroup V_{F} := Gal(M/F) is isomorphic to a Klein 4-group. The dihedral group of order 8 contains two distinct, non-conjugate subgroups which are isomorphic to the Klein 4-group. This is most easily seen by viewing D_{8} as the symmetry group of a square, as in the figure below, in which V_{F} is identified with the subgroup generated by the reflections about the two diagonals. These two reflections can be labeled as τ_{M} and τ′_{M}, in such a way that

M^{V_{F}} = F,  M^{τ_{M}} = M,  M^{τ′_{M}} = M′.

The second Klein four-group, which shall be denoted V_{K}, is generated by the reflections about the vertical and horizontal axes of symmetry of the

17

square. We label these reflections as τ_{L} and τ′_{L}, as shown in the figure below.

τ′_{M} _{⟳}  _{τ_{L}}_{⟳}  _{τ_{M}}_{⟲} •  ∘  •  _{τ′_{L}}_{⟲} ∘  ∘ •  ∘  •  (20)

Now let

K := M^{V_{K}} ,  L := M^{τ_{L}},  L′ = M^{τ′_{L}}.

These fields fit into the following diagram of field extensions, where each unbroken line indicates an extension of degree 2:

[figure: M; M; M′; L′; L; F; K; ℚ]  (21)

Let

χ_{M}, χ′_{M} : G_{F} ⟶ {±1}

denote the Galois characters of the real quadratic field F which cut out the extensions M and M′, and let

χ_{L}, χ′_{L} : G_{K} ⟶ {±1}

be the quadratic characters of the imaginary quadratic field K which cut out the extensions L and L′. We will often view these characters as idèle class characters defined on 𝔸_{F}^{×} and on 𝔸_{K}^{×} respectively. Finally, let ε_{F} and ε_{K} denote the quadratic Dirichlet characters attached to F and K, and let N_{ℚ}^{F} : 𝔸_{F}^{×} ⟶ 𝔸_{ℚ}^{×} and N_{ℚ}^{K} : 𝔸_{K}^{×} ⟶ 𝔸_{ℚ}^{×} denote the norms on adèles.

18

Proposition 3.1.

(1) The field K is a quadratic imaginary field.

(2) The characters χ_{M}, χ′_{M}, χ_{L} and χ′_{L}, viewed as idèle class characters of F and K respectively, satisfy

χ_{M}χ′_{M} = ε_{K} ∘ N_{ℚ}^{F}; χ_{L}χ′_{L} = ε_{F} ∘ N_{ℚ}^{K}.

(3) The central character of χ_{M} and χ′_{M} is ε_{K}, and the central character of χ_{L} and χ′_{L} is ε_{F}.

(4) The following two-dimensional representations of G_{ℚ} are isomorphic:

Ind_{F}^{ℚ}χ_{M} = Ind_{F}^{ℚ}χ′_{M} = Ind_{K}^{ℚ}χ_{L} = Ind_{K}^{ℚ}χ′_{L}.

Proof. The quadratic field K is of the form ℚ(\sqrt{−d}), where −d is defined (modulo squares in ℚ^{×}) by

−d = N_{F/ℚ}(α), with M = F (\sqrt{α}).

The fact that M is ATR implies that −d is a negative rational number, and therefore that K is an imaginary quadratic field. The second part follows directly from the field diagram (21) above. To prove the third part, note that part (2) implies that the central character of L restricted to the group of norms from K is equal to ε_{F}. (This is because χ_{L}(x) = χ′_{L}(x̄), where x ↦ x̄ is complex conjugation). Class field theory implies that this central character differs from ε_{F} by a power of ε_{K}. But the central character of χ_{L} cannot be ε_{F} ε_{K} since this is an odd Dirichlet character and the central character of a finite order Hecke character of an imaginary quadratic field is necessarily even, because the map from the group of components of ℝ^{×} to the group of components of ℂ^{×} is trivial. Finally, the proof of part (4) is a simple exercise in representation theory: the four representations that are listed in (4) are all isomorphic to the unique irreducible two-dimensional representation of Gal(M/ℚ).

As at the end of the previous section §3.1, let E be a ℚ-curve over a real quadratic field F and A = Res_{F/ℚ}(E), let f = f_{E} ∈ S_{2}(Γ_{ε}(N)) denote the modular form associated to it and let K_{f} denote the imaginary quadratic field generated over ℚ by the Fourier coefficients of f.

Theorem 3.1 (Tian, Yuan, Zhang and Zhang). Let K be a quadratic imaginary field satisfying the Heegner hypothesis, and let χ : 𝔸_{K}^{×} ⟶ ℂ^{×} be a finite order Hecke character of K satisfying

χ|_{𝔸_{ℚ}^{×}} = ε_{f}^{−1}.  (22)

19

Then

(i) The L-function L(f/K, χ, s) vanishes to odd order at s = 1; (ii) If L′(f/K, χ, 1) ≠ 0, then (A(K^{ab}) ⊗ ℂ)^{χ} has rank one over K_{f} ⊗_{ℚ} ℂ,

and LLI(A/K^{ab})^{χ} is finite.

Proof. The modular form f gives rise to a cuspidal automorphic representation π of GL_{2}(𝔸_{ℚ}) whose central character is ω_{π} = ε_{f}, the nebentypus of f. Condition (22) ensures that the tensor product of the motives attached to π and χ is self-dual, and therefore the L-function

L(f/K, χ, s) = L(π, χ, s − \frac{1}{2})

satisfies a functional equation whose central critical point is s = 1; since the discriminant of K is relatively prime to N, the sign of this functional equation is (−1)^{♯Σ}, where

Σ = {primes ℓ inert in K such that ord_{ℓ}(N) is odd} ∪ {∞}.

The Heegner hypothesis satisfied by K says that Σ = {∞} and thus the sign is −1; this implies (i).

As a consequence of (13), the complex vector space (A(K^{ab}) ⊗ ℂ)^{χ} is naturally a K_{f} ⊗_{ℚ} ℂ-module. Part (ii) is a theorem of Tian-Zhang^{39} which follows as a corollary of [42, Theorem 1.3.1] by applying Kolyvagin’s method. Since the reference^{39} is not currently available, the reader may consult [42, Theorem 1.4.1] and, for the precise statement quoted here, [44, Theorem 4.3.1].

We are now ready to prove Theorem 1.4 of the introduction.

Theorem 3.2. Let E be a ℚ-curve over F of square conductor 𝔑, and let M/F be an ATR extension of F of discriminant prime to 𝔑. If L′(E_{M}/F, 1) ≠ 0, then E_{M}(F) has rank one and LLI(E_{M}/F) is finite.

Proof. By (11) and the Artin formalism for L-series,

L(E_{M}/F, s) = L(E, χ_{M}, s) = L(f ⊗ χ_{M}/F, s) = L(f ⊗ Ind_{F}^{ℚ}χ_{M}, s). (23)

By part 4 of Proposition 3.1,

L(f ⊗ Ind_{F}^{ℚ}χ_{M}, s) = L(f ⊗ Ind_{K}^{ℚ}χ_{L}, s) = L(f ⊗ χ_{L}/K, s). (24)

It follows from (23) and (24) that

L′(E_{M}/F, 1) = L′(f ⊗ χ_{L}/K, 1) = L′(f^{σ} ⊗ χ_{L}/K, 1).  (25)

20

Therefore the two rightmost expressions in (25) are non-zero by assumption, so that the product

L(f ⊗ χ_{L}/K, s)L(f^{σ} ⊗ χ_{L}/K, s) = L(A/K, χ_{L}, s)

vanishes to order exactly 2 = [K_{f} : ℚ] at s = 1. By Theorem 3.1, it follows that A(L)^{−} is of rank two, where A(L)^{−} denotes the subgroup of the Mordell-Weil group of A(L) of points whose trace to K is trivial. In particular, the Galois representation Ind_{K}^{ℚ}χ_{L} occurs in A(\overline{ℚ}) ⊗ ℂ with multiplicity 2. Hence, invoking once again part 4 of Proposition 3.1, we find that

rank(A(M)^{−}) = 2.

But since M contains F and since A is isogenous over F to E ×E, it follows that

rank(E(M)^{−}) = rank(E_{M}(F)) = 1.

The result about the ranks follows. The result about the finiteness of LLI(E_{M}/F) follows in the same way from part (ii) of Theorem 3.1.

4. Heegner points on Shimura’s elliptic curves

Implicit in the proof of Theorem 3.2 (via the use that is made of it in the proof of Theorem 3.1) is the construction of a Heegner point P_{M} ∈ E_{M}(F) arising from the image of certain CM divisors on X_{ε}(M) via the modular parametrisation (10). We now wish to make this construction explicit in the case where the ℚ-curve E has everywhere good reduction over the real quadratic field F. The ℚ-curves with this property are sometimes called Shimura elliptic curves because they were first systematically considered by Shimura. More precisely, it is shown in Shimura^{35} how to associate a Shimura elliptic curve over F = ℚ(\sqrt{N}) to any classical elliptic modular form f ∈ S_{2}(Γ_{0}(N), ε_{N}) with quadratic fourier coefficients. (Cf. also (17).)

It will be assumed throughout this chapter that E/F is a Shimura elliptic curve, and that f is the corresponding elliptic modular form. We also assume for simplicity that N is odd, and thus square-free.

Remark 4.1. According to calculations performed by the third author using PARI^{29} (extending the data gathered in [6, §6], Pinch^{31} in the range N ≤ 1000), there exists an eigenform f ∈ S_{2}(Γ_{0}(N), ε_{N}) of prime level 1 < N < 5000, with fourier coefficients in a quadratic imaginary extension K_{f} = ℚ(\sqrt{−d}) for precisely the values of N and d listed in the following table.

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N 29  37  41 109 157 229 257 337 349
d  5  1  2  3  1  5  2  2  5
N 373 397 421 461 509 877 881 997 1069
d 1 1 7 5 5 1 2 3 1
N 1709 1861 2657 4481 4597
d  5  5  2  11  1

Associated to each such eigenform there is a Shimura elliptic curve over F = ℚ(\sqrt{N}). Furthermore, according to computations due to Cremona, Dembelé, Elkies and Pinch, there are only four primes N in the range [1, 1000] for which there exists an elliptic curve with good reduction everywhere over F = ℚ(\sqrt{N}) which is not a ℚ-curve, namely, N = 509, 853, 929 and 997. It is hard to predict whether the preponderance of Shimura elliptic curves among elliptic curves of conductor one will persist or is merely an artefact of the relatively low ranges in which numerical data has been gathered. Note that it is not even known whether there exist infinitely many Shimura elliptic curves over real quadratic fields, while it is a theorem of S. Comalada^{5} that there are infinitely many elliptic curves over real quadratic fields with good reduction everywhere.

4.1. An explicit Heegner point construction

Let us recall the diagram of field extensions introduced in (21):

[figure: M; M; M′; ℚ(\sqrt{N}, \sqrt{−d}); L′; L; F; ℚ(\sqrt{−Nd}); K; ℚ]  (26)

The following lemma is crucial in constructing the point P_{M} ∈ E(M)^{−} = E_{M}(F) explicitly.

Lemma 4.1. Let M be an ATR extension of F and let K be the quadratic imaginary field attached to M as in the diagram (26). Then K has a

22

(canonical) ideal N of K of norm N. In particular, all the prime divisors of N are either split or ramified in K.

Proof. The conductor-discriminant formula combined with part 4 of Proposition 3.1 show that

disc(F)Nm_{F/ℚ}(disc(M/F)) = disc(K)Nm_{K/ℚ}(disc(L/K)).

Therefore, after setting

N_{ram} = gcd(N, disc(K)), N_{split} = N/N_{ram},
N_{ram} = (N_{ram}, \sqrt{disc(K)}), N_{split} = (N_{split}, disc(L/K)),

we find that N := N_{ram}N_{split} gives the desired ideal of norm N.

Let 𝔸_{K} denote the ring of adèles of K, and let

Ô_{K}^{×} := ∏_{v} O_{v}^{×}

denote the maximal compact subgroup of the group 𝔸_{K,fin}^{×} of finite idèles of K. Given a rational integer c ≥ 1, (c, N) = 1, we define

U_{c} = ℤ̂^{×}(1 + cÔ_{K})ℂ^{×} ⊂ 𝔸_{K}^{×}.

By class field theory, the quotient G_{c} := 𝔸_{K}^{×}/(K^{×}U_{c}) is identified with Gal(H_{c}/K), where H_{c} is the ring class field of K of conductor c.

As a piece of notation, we shall write H_{c} for the ring class field attached to the order in K of conductor c ≥ 1 and write K_{𝔞} for the ray class field of conductor 𝔞.

Define

U_{c}^{+} = {β ∈ U_{c} such that (β)_{N} ∈ ker(ε) ⊂ (ℤ/Nℤ)^{×}},
U_{c}^{−} = \{β ∈ U_{c} such that (β)\overline{N} ∈ ker(ε) ⊂ (ℤ/Nℤ)^{×}\},

and Ũ_{c} = U_{c}^{+} ∩ U_{c}^{−}. Here (β)_{N} denotes the image of the local term of the idèle β at N in the quotient O_{N}^{×} /(1 + N · O_{N}) ≃ (ℤ/Nℤ)^{×}. Similarly for \overline{N}. This way we can regard the character ε as having source either O_{N}^{×} or

O× ; \overline{N}.

Set

G̃_{c} := 𝔸_{K}^{×}/(K^{×}Ũ_{c}) = Gal(H̃_{c}/K),

where H̃_{c} is a biquadratic extension of the ring class field H_{c}. It can be written as H̃_{c} = L_{c}L′_{c}, where L_{c} (resp. L′_{c}) is the class field attached to U_{c}^{+} (resp. U_{c}^{−}).

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Proposition 4.1. The relative discriminant of L/K factors as d(L/K) = c · N, where c is a positive integer such that L ⊂ L_{c} and L′ ⊂ L′_{c} and thus M ⊂ H̃_{c}.

More precisely, we have c = 2^{t} · c_{0} where 0 ≤ t ≤ 3 and c_{0} is odd and square-free. The proof of this proposition is an exercise in class field theory, which we relegate to §4.4 for the convenience of the reader.

We now explain how to construct a degree zero divisor on X_{ε}(N) defined over H̃_{c}. To do this, let A_{c} be an elliptic curve satisfying

End(A_{c}) = O_{c},

where O_{c} := ℤ + cO_{K} is the order in K of conductor c. Such a curve, along with its endomorphisms, may be defined over the ring class field H_{c}. The module A_{c}[N] of N -torsion points is therefore defined over H_{c}, yielding a point P_{c} := [A_{c}, A_{c}[N]] ∈ X_{0}(N)(H_{c}).

The action of G_{H_{c}} := Gal(\overline{ℚ}/H_{c}) on the points of this group scheme gives a Galois representation

ρ_{N} : G_{H_{c}} ⟶ (ℤ/Nℤ)^{×}.

The composition of ρ_{N} with the nebentypus character ε is a quadratic character of G_{H_{c}}, which cuts out the quadratic extension L_{c} of H_{c}. The point P_{c} lifts to two points P_{c}^{+} and P_{c}^{−} in X_{ε}(N)(L_{c}) which are interchanged by the action of Gal(L_{c}/H_{c}); we do not specify the order in which these points are to be taken. Similarly, we can replace the module A_{c}[N] by A_{c}[\overline{N}], mimic the above construction and obtain points P_{c}^{′+} and P_{c}^{′−} defined over L′_{c}.

Definition 4.1. Let

CM(c) = ⋃{P_{c}^{+}, P_{c}^{−}, P_{c}^{′+}, P_{c}^{′−}} ⊂ X_{ε}(N)(H̃_{c})

be the set of Heegner points on X_{ε}(N) obtained by letting A_{c} run over all isomorphism classes of elliptic curves with CM by O_{c}.

If we let h(O_{c}) denote the cardinality of the group Pic(O_{c}) of classes of locally free ideals of O_{c}, the cardinality of CM(c) is 4h(O_{c}). In fact, CM(c) is naturally the disjoint union of the two subsets CM(c) ∩ X_{ε}(N)(L_{c}) and CM(c) ∩ X_{ε}(N)(L′_{c}), each of cardinality 2h(O_{c}).

A Heegner point P ∈ CM(c) of conductor c may be described by a triple ([𝔞], 𝔫, t), where

• [𝔞] ∈ Pic(O_{c}) is the class of an invertible O_{c}-module of K,

24

• 𝔫 is an integral ideal of O_{c} such that the quotient O_{c}/𝔫 is cyclic of order N,

• t is an orbit for the action of ker(ε) of an element of order N in 𝔞𝔫^{−1}/𝔞 ≅ ℤ/Nℤ.

Let C̃ be the quotient of the ray class group of K of conductor cN for which Artin’s reciprocity map of global class field theory furnishes a canonical isomorphism

rec : C̃ ⟶^{∼} Gal(H̃_{c}/K).

Let O = O_{c} denote the order of conductor c in K. There are natural exact sequences, sitting in the commutative diagram

\xymatrix{1 \ar[r] & Gal(H̃_{c}/H_{c}) \ar[r] \ar[d]^{rec} & Gal(H̃_{c}/K) \ar[d]^{rec} \ar[r]^{^{res^{H̃c/Hc}}} & Gal(H_{c}/K) \ar[r] \ar[d]^{rec} & 1 \\ 1 \ar[r] & ⟨[β_{0}], [β′_{0}]⟩ \ar[r] & C̃ \ar[r] & Pic(O) \ar[r] & 1,}

where the vertical arrows are isomorphisms. Here, β_{0} ∈ O_{N}^{×} and β′_{0} ∈ O× ; \overline{N} are elements such that ε(β_{0}) = −1 and ε(β′_{0}) = −1. Artin’s reciprocity map induces an isomorphism

Gal(H̃_{c}/H_{c}) ≃ O_{N}^{×} / ker(ε) × O× ; \overline{N} / ker(ε) ≃ ℤ/2ℤ × ℤ/2ℤ.

We thus can formally write elements of C̃ as classes of enhanced ideals, which are defined as elements of the form \underline{𝔟} := β_{N} β\overline{N} ∏_{℘∤N} ℘^{n_{℘}}, taken up to principal ideals (b) with b ∈ K^{×}. Here β_{N} and β\overline{N} belong to K_{N}^{×}/ ker(ε) and K× ; \overline{N}/ ker(ε) respectively, ℘ runs over all prime invertible ideals of O not dividing N, and the exponents n_{℘} are integers which are almost all zero. We say an enhanced ideal is integral if β_{N} and β\overline{N} have representatives in O_{N}^{×} and O× ; \overline{N} respectively, and n_{℘} ≥ 0 for all ℘. The image of the class \underline{𝔟} in

Pic(O) is simply the class of the ideal 𝔟 = N^{ord_{N} (β_{N})}\overline{N}^{ord\overline{N} (β\overline{N})} ∏_{℘∤N} ℘^{n_{℘}} generated by it.

By Shimura’s reciprocity law,

rec(\underline{𝔟})(D) = \underline{𝔟}^{−1} ⋆ D  (27)

for all \underline{𝔟} ∈ C̃ and all divisors D ∈ J_{ε}(N)(H̃_{c}) supported on CM(c).

On the left hand side we make use of the natural Galois action of Gal(H̃_{c}/K) on J_{ε}(N)(H̃_{c}), via Artin’s reciprocity isomorphism. On the right hand side, a class [\underline{𝔟}] ∈ C̃ acts on CM(c) by the rule

\underline{𝔟} ⋆ P = ([𝔞𝔟^{−1}], 𝔫, φ_{𝔟}(β_{N} t)),  (28)

25

where P = ([𝔞], 𝔫, t) ∈ CM(c), \underline{𝔟} = β_{N} β\overline{N} ∏_{℘∤N} ℘^{n_{℘}} is an integral representative of its class and φ_{𝔟} : ℂ/𝔞 → ℂ/𝔞𝔟^{−1} is the natural projection map. Writing P = [τ] ∈ X_{ε}(N)(ℂ) for some τ ∈ H, let γ\underline{𝔟} ∈ GL_{2}^{+}(ℚ) be such that \underline{𝔟} ⋆ P = [γ\underline{𝔟}τ].

Besides this action, there is also the diamond involution W_{ε}, acting on P = [τ] ∈ X_{ε}(N)(ℂ) as W_{ε}([τ]) = [γ_{ϵ}τ] and on P = ([𝔞], 𝔫, t) ∈ CM(c) as

W_{ε}(P) = ([𝔞], 𝔫, dt), for γ_{ϵ} = ( \begin{matrix}a & b \\ Nc & d \end{matrix} ) ∈ Γ_{0}(N) \ Γ_{ε}(N).  (29)

The cardinality of CM(c) is 4h(O) and it is acted on freely and transitively by the group ⟨W_{N}, W_{ε}⟩ × C̃_{M}, where we let C̃_{M} := rec^{−1}(Gal(H̃_{c}/M)) ⊂ C̃_{K}. Note that the restriction map res_{H̃_{c}/H_{c}} induces an isomorphism C̃_{M} ≅ Pic(O) ≅ Gal(H_{c}/K).

It is our aim now to define a point P_{M} ∈ E(M) (and thus also, by conjugation over F, a point P_{M′} ∈ E(M′)) on the elliptic curve E, rational over the ATR extension M/F. We shall construct P_{M} as a suitable linear combination of certain points P_{L} ∈ A(L) and P_{L′} ∈ A(L′) on the abelian surface A = Res_{F/ℚ}(E). These points are defined as the trace to L of the projection of P_{c}^{+} ∈ X_{ε}(N)(L_{c}) (respectively of P_{c}^{′+} ∈ X_{ε}(N)(L′_{c})) on A.

Before doing so, we first observe that choosing P_{c}^{−} = W_{ε}(P_{c}^{+}) instead of P_{c}^{+} (and similarly P_{c}^{′−} instead of P_{c}^{′+}) is unimportant for our construction, as the next lemma shows that both lead to the same point on A up to sign and torsion. Recall the canonical projection π_{f} : J_{ε}(N) → A defined over ℚ and reviewed in (19), which can be composed with the natural embedding of X_{ε}(N) into its jacobian J_{ε}(N) given by the map P ↦ P − i∞. By an abuse of notation, we continue to denote by π_{f} this composition.

Lemma 4.2. For any P ∈ X_{ε}(N)(\overline{ℚ}), the point π_{f}(P)+π_{f}(W_{ε}(P)) belongs to A(F)_{tors}.

Proof. There is a natural decomposition S_{2}(Γ_{ε}(N)) = S_{2}(Γ_{0}(N)) ⊕ S_{2}(Γ_{0}(N), ε) corresponding to the eigenspaces of eigenvalue ±1 with respect to the action of the involution W_{ε}. The rule f(z) ↦ f(z)dz yields an identification of S_{2}(Γ_{ε}(N)) with the space of holomorphic differentials on X_{ε}(N)_{ℂ}. Via this isomorphism, π_{f}^{∗}H^{0}(Ω_{A}^{1}) is contained in S_{2}(Γ_{0}(N), ε). Consequently, π_{f}(P − i∞) = −π_{f}(W_{ε}(P − i∞)) and

π_{f}(P) + π_{f}(W_{ε}(P)) = π_{f}(P − i∞) + π_{f}(W_{ε}(P) − i∞)
= π_{f}(P − i∞) + π_{f}(W_{ε}(P) − W_{ε}(i∞)) + π_{f}(W_{ε}(i∞) − i∞)
= π_{f}(W_{ε}(i∞) − i∞).

26

This last expression is a torsion point on A(F) by the Manin-Drinfeld theorem which asserts that degree zero cuspidal divisors on a modular curve give rise to torsion elements in its Jacobian.

We now set

P_{L} = Tr_{L_{c}/L}(π_{f}(P_{c}^{+})) ∈ A(L).

Note that τ_{M}(P_{L}^{+}) is either equal to Tr_{L′_{c}/L′}(π_{f}(P′_{c}^{+})) or to Tr_{L′_{c}/L′}(π_{f}(P′_{c}^{−})). Without loss of generality, assume that τ_{M}(P_{L}^{+}) = Tr_{L′_{c}/L′}(π_{f}(P′_{c}^{+})) and denote it by P′_{L}.

Set

u = \{ \begin{matrix}2 & if K = ℚ(\sqrt{−1}) and c = 1; \\ 3 & if K = ℚ(\sqrt{−3}) and c = 1; \\ 1 & otherwise, \end{matrix}

and define

P_{A,M} := \frac{1}{u}(P_{L} + P′_{L}), P_{A,M′} := \frac{1}{u}(P_{L} − P′_{L}).

The construction of the point P_{A,M} is illustrated in the figure below.

τ′_{M} _{⟳}  _{τ_{L}}_{⟳}  _{τ_{M}}_{⟲} •_{P_{A,M′}}  _{P_{A,M}}_{P_{L}} ∘  •_{−P′_{L}}  _{τ′_{L}}_{⟲}_{P′_{L}} ∘  ∘ •_{−P_{A,M}}  _{−P_{L}} _{−P_{A,M′}} ∘  •

This figure suggests–and it is indeed easy to check–that

P_{A,M} ∈ A(M), P_{A,M′} ∈ A(M′).

Recall that the morphism φ_{F} : A_{F} ⟶ E introduced in (19) is defined over F ⊂ M, and therefore that the point

P_{M} := φ_{F} (P_{A,M})

belongs to E(M). As a by-product of our explicit construction we obtain the following analytic formula for calculating the point P_{M}.

27

Theorem 4.1. Let τ_{c}, τ′_{c} ∈ H be elements representing the Heegner points P_{c}^{+}, P_{c}^{′+} ∈ X_{ε}(N)(H̃_{c}). Set

z_{M} = ∑_{\underline{𝔟}∈C̃_{M}}[∫_{i∞}^{γ\underline{𝔟}τ_{c}} (f_{E}(τ) + f_{E}|_{W_{N}} (τ)) dτ + ∫_{i∞}^{γ\underline{𝔟}τ′_{c}} (f_{E}(τ) + f_{E}|_{W_{N}} (τ)) dτ] .

(30)

Then P_{M} = η(z_{M}) where η is the Weierstrass parametrization

η : ℂ/Λ_{E} ⟶ E(ℂ), η(z) = (℘(z), ℘′(z)).  (31)

Here, ℘ is the Weierstrass function associated with the lattice of periods

Λ_{E} := {∫_{δ}(f_{E} + f_{E}|_{W_{N}})dτ}

where δ ∈ H_{1}(X_{ε}(ℂ), ℤ) runs over the cycles of X_{ε}(ℂ) such that ∫_{δ}(f_{E} − f_{E}|_{W_{N}})dτ = 0.

4.2. Heegner points and ATR cycles

The main conjecture that will be formulated in this section relates the Heegner point P_{M} with the Stark Heegner point P_{M}^{?} arising from ATR cycles. Recall that Gal(F/ℚ) = {1, τ} and D_{F} = disc(F). Let also c_{E/F} (resp. c_{E^{τ} /F}) denote either the real period or twice the real period of E/ℝ (resp. of E^{τ} /ℝ) depending on whether E(ℝ) (resp. E^{τ} (ℝ)) is connected or not.

Conjecture 4.1. The ATR point P_{M}^{?} is of infinite order if and only if P_{M} is of infinite order and L(E/F, 1) ≠ 0. More precisely,

P_{M}^{?} = 2^{s}ℓ · P_{M}  (32)

where ℓ ∈ ℚ^{×}, which depends only on (E, F) and not on M, satisfies

ℓ^{2} = \frac{L(E/F, 1)}{Ω_{E/F}}, with Ω_{E/F} = \frac{c_{E/F} · c_{E^{τ} /F}}{D_{F}^{1/2} · |E_{tor}(F)|^{2}} ,

and s ∈ ℤ depends on M.

Below we collect some numerical data in support of conjecture 4.1. Besides the numerical evidence, Conjecture 4.1 is also motivated by the equality

L′(E/M, 1) = L(E/F, 1)L′(E/F, χ_{M}, 1)

as recalled in the proof of Theorem 3.2, and by the facts that

28

(1) the Néron-Tate height of the Stark-Heegner point P_{M}^{?} is expected^{8} to be related in a simple way to L′(E/M, 1).

(2) the extension of the Gross-Zagier formula^{42} proved by Yuan, Zhang and Zhang should in principle lead to an analogous relationship between the Heegner point P_{M} and the derivative L′(f/K, χ_{L}, 1) = L′(f/F, χ_{M}, 1) = L′(E/F, χ_{M}, 1).

It would be interesting to formulate a precise recipe predicting the power of 2 that arises as a fudge factor in (32). The authors have not made a serious attempt to do this.

4.3. Numerical examples

For N = 29, 37, 41 it is known (cf. Shiota^{36} and [6, § 6]) that there is a unique Shimura elliptic curve defined over F = ℚ(\sqrt{N}) up to isogeny over F.

The aim of this section is to provide numerical evidence for conjecture 4.1, which we have gathered by explicitly computing the points P_{M} and P_{M}^{?} for several ATR extensions M/F on each of these three elliptic curves. The computation of the Heegner point P_{M} was performed with the software package PARI^{29} by exploiting formula (30) and the material in Shiota^{36} and [6, § 6] to produce a complex invariant z_{M} ∈ ℂ/Λ_{E} mapping to P_{M} under the Weierstrass uniformisation. Similarly, the ATR point P_{M}^{?} was computed by following the method explained in [8, Section 3].

In fact, for our experiments it was sufficient to compute the element z_{M}^{?} ∈ ℂ/Λ_{E} mapping to P_{M}^{?} under the Weierstrass uniformisation. For several values of M, the invariants z_{M} and z_{M}^{?} were calculated to roughly 50 digits of decimal accuracy, and the constants s and ℓ in (32) could then be obtained by picking a basis (e_{1}, e_{2}) for Λ_{E} and searching for a linear dependence relation with small integer coefficients between the four complex numbers z_{M}, z_{M}^{?}, e_{1} and e_{2}, using Pari’s lindep command.

This approach represents a dramatic improvement over the one that had to be followed in [8, Section 3] in which only the point P_{M}^{?} was computed. In practice, recognizing P_{M}^{?} as an algebraic point using standard rational recognition programs is difficult once the height of P_{M}^{?} becomes large. The strategy followed in [8, Section 3] consists in performing an independent search for a generator of E(M)–a computationally difficult and time-consuming task–in order to check that P_{M}^{?} indeed agreed with a point of small height on E(M) to within the calculated decimal accuracy. The new approach based on the Heegner point P_{M} makes the experimental ver-

29

ifications of the conjectures of [8, §2] much more systematic and efficient, and allows them to be carried out for much further ranges.

In the tables below, we have followed almost the same notations as in [8, §3]. In particular, we have written M = F (β) with β^{2} ∈ F, and denoted by D_{M} = Nm_{F/ℚ}(disc(M/F)) the absolute discriminant. (Note that K is used in Darmon-Logan^{8} to denote the field that we call M in the present work.) As before, L/K denotes the quadratic extension sitting in the Galois closure of M as in the field diagram (26), and we denote D_{K} = disc(K/ℚ) and D_{L} = Nm_{K/ℚ}(disc(L/K)) and ℓ^{2} ∈ ℚ^{×} is the value we found numerically for the constant alluded to in Conjecture 4.1. Finally note that ℓ and s uniquely determine P_{M} up to sign and E(M)_{tor}.

The case N = 29. Let δ = 2 + ω = (5 + \sqrt{29})/2. Shiota’s Weierstrass equation for E_{N} is given by

E_{29} : y^{2} + xy + δ^{2}y = x^{3},

whose discriminant is Δ_{29} = −δ^{10}. Our calculations convincingly suggest that

ℓ^{2} = \frac{L(E_{29}/F, 1)}{Ω_{E_{29}/F}} = 1

and that the point P_{M}^{?} and s are given in the following table. The table suggests that s = −2 in all cases that have been calculated for this particular curve.

DM = DK · c2  β2  DL  |Pic(Oc)|  P? M  s
−7 = −7 · 1  −1 + ω  29  1  (β2 + 3, − \frac{5}{2} β3 − 3β2 − 8β − \frac{19}{2} )  −2
−16 = −4 · 22  2 + ω  22 · 29  1  ( \frac{β2}{2} , − \frac{5}{4} β3 − \frac{11}{4} β2 − \frac{β}{4} − \frac{1}{2} )  −2
−23 = −23 · 1  17 + 8ω  29  3  ( \frac{1}{8} (11β2 + 5), − \frac{13}{8} β3 − β2 − \frac{7}{8} β − \frac{1}{2} )  −2
−351 = −35 · 1  19 + 9ω  29  2  ( \frac{1}{5} (2β2 + 1), − \frac{59}{225} β3 − \frac{43}{90} β2 − \frac{89}{450} β − \frac{29}{90} )  −2
−352 = −35 · 1  4 + 3ω  29  2  (− \frac{1}{15} (4β2 + 11), − \frac{1}{150} (17β3 + 105β2 + 43β + 270))  −2
−59 = −59 · 1  61 + 28ω  29  3  (− \frac{1}{9} , − \frac{11}{1512} β3 − \frac{5}{56} β2 − \frac{1}{1512} β + \frac{1}{504} )  −2
−63 = −7 · 32  3ω  32 · 29  4  ( \frac{7}{9} β2 + 5, \frac{26}{27} β3 − \frac{11}{9} β2 + \frac{57}{9} β − 8)  −2
−64 = −4 · 42  4 + 2ω  24 · 29  2  (− \frac{1}{4} , − \frac{3}{8} β3 − \frac{5}{4} β2 − \frac{β}{4} − \frac{3}{8} )  −2
−80 = −20 · 22  1 + ω  22 · 29  4  ( \frac{1}{10} (43β2 + 51), − \frac{517}{50} β3 − \frac{93}{20} β2 − \frac{1233}{100} β − \frac{111}{20} )  −2
−91 = −91 · 1  7 + 5ω  29  2  ( \frac{1}{13} (98β2 + 387), − \frac{18939}{845} β3 − \frac{111}{26} β2 − \frac{150109}{1690} β − \frac{439}{26} )  −2
−175 = −7 · 52  −5 + 5ω  52 · 29  6  (− \frac{6}{50} β2 − 2, \frac{1}{10} β3 − \frac{11}{25} β2 + \frac{98}{100} β − \frac{45}{10} )  −2

Table 2: ATR extensions of ℚ(\sqrt{29}) and ATR points on E29

The case N = 37. Letting ω = \frac{1+\sqrt{37}}{2}, Shiota’s Weierstrass equation for

30

E_{37} is given by

E_{37} : y^{2} + y = x^{3} + 2x^{2} − (19 + 8ω)x + (28 + 11ω),

and its discriminant is Δ_{37} = (5 + 2ω)^{6}. Note that 5 + 2ω is a fundamental unit of F. Our calculations are consistent with the fact that

ℓ^{2} = \frac{L(E_{37}/F, 1)}{Ω_{E_{37}/F}} = 1.

More precisely, the Stark-Heegner point P_{M}^{?} and s are given in the tables below.

DM = DK · c2  β2  DL  |Pic(Oc)|  P? M  s
−3 = −3 · 1  −3 + ω  37  1  (− \frac{2}{3} β − \frac{13}{3} , − \frac{61}{18} β3 − \frac{169}{9} β − \frac{1}{2} )  −1
−7 = −7 · 1  1 + ω  37  1  ( \frac{2}{7} β − \frac{3}{7} , − \frac{57}{98} β3 − \frac{44}{49} β − \frac{1}{2} )  −1
−11 = −11 · 1  38 + 15ω  37  1  (− \frac{2}{165} β2 − \frac{104}{165} , − \frac{17}{1210} β3 − \frac{2}{605} β − \frac{1}{2} )  −1
−16 = −4 · 22  5 + 2ω  22 · 37  1  ( \frac{β2}{8} − \frac{5}{8} , \frac{β3}{8} − \frac{1}{2} )  −2
−48 = −3 · 42  2 + ω  42 · 37  3  ( \frac{115}{588} β2 − \frac{80}{147} , − \frac{11225}{24696} β3 − \frac{1529}{6174} β − \frac{1}{2} )  −1
−64 = −4 · 42  10 + 4ω  42 · 37  2  (− \frac{β2}{8} − \frac{3}{4} , − \frac{β3}{8} − \frac{1}{2} )  −2
−67 = −67 · 1  193 + 76ω  67  1  (−1, − \frac{1}{2} + \frac{1}{2} β)  −2
−75 = −3 · 52  −15 + 5ω  52 · 37  3  ( \frac{196}{775} β2 + \frac{136}{27} , − \frac{1559}{12150} β3 − \frac{25732}{6075} β − 1/2)  −1
−192 = −3 · 82  18 + 8ω  82 · 37  6  ( \frac{7}{3} + \frac{7}{6} ω, − \frac{1}{2} + \frac{1}{36} ( \frac{85}{3} + \frac{14}{3} \sqrt{37})β)  −2
−275 = −11 · 52  445 + 180ω  52 · 37  4  ( \frac{2}{11} + \frac{4}{11} ω, − \frac{1}{2} + \frac{1}{242} ( \frac{62}{7} + \frac{9}{7} \sqrt{37})β)  −2
−448 = −7 · 82  2 + 2ω  82 · 37  4  ( \frac{45}{7} + \frac{39}{14} ω, − \frac{1}{2} + \frac{1}{196} ( \frac{689}{2} \sqrt{37} + \frac{4191}{2} )β)  −2

Table 3: ATR extensions of ℚ(\sqrt{37}) and ATR points on E37

The case N = 41. Shiota’s Weierstrass equation for E_{41} is

E_{41} : y^{2} = x^{3} − \frac{17}{48}x + (−\frac{5}{32} + \frac{1}{27}\sqrt{41})

The computations in [8, Section 3] used instead curve E′_{41} : y^{2} + xy = x^{3} − (32 + 5\sqrt{41})x. This Weierstrass equation was first found by Oort, and there is an explicit isogeny ψ : E′_{41} → E_{41} of degree 2. Following the approach in [8, Section 3], points P_{M}^{?} listed below are points on E′_{41}. Since the isogeny ψ is explicit, it is an easy task to transfer them to points on E_{41}, and this is what we did in order to compare the Heegner points P_{M} ∈ E_{41}(M) with the Stark-Heegner points ψ(P_{M}^{?}) ∈ E_{41}(ℂ). In this case, calculations suggest once again that

ℓ^{2} = \frac{L(E_{41}/F, 1)}{Ω_{E_{41}/F}} = 1,

31

while the values of the exponent s also appear in the table below.

DM = DK · c2  β2  DL  |Pic(Oc)|  P? M  s
−4 = −4 · 1  27 + 10ω  41  1  (− \frac{1}{4} , − \frac{β}{2} + \frac{1}{8} )  1
−8 = −8 · 1  −248 + 67ω  41  1  (− \frac{1}{268} (3β2 + 1481), \frac{1}{536} (−254β3 + 3β2 − 108954β + 1481))  0
−20 = −20 · 1  697 + 258ω  41  2  ( \frac{1}{43} (β2 − 9), \frac{1}{258} (−β3 − 3β2 + 181β + 27))  0
−23 = −23 · 1  398 + 144ω  41  3  ( \frac{−71027β2−1271153}{9884736} , \frac{−1095348β3+9304537β2+16459332β+166521043}{2589800832} )  0
−32 = −8 · 22  1 + ω  22 · 41  2  ( \frac{29β2+49}{4} , \frac{1}{16} (−359β3 − 58β2 − 611β − 98))  0
−36 = −4 · 32  6 + 3ω  32 · 41  4  (−8 + 2ω, ( \frac{7}{2} − \frac{1}{2} \sqrt{41})(1 + 5β))  −1
−40 = −40 · 1  35 + 13ω  41  2  (9 + \frac{27}{8} ω, − \frac{171}{32} − \frac{27}{32} \sqrt{41} + \frac{3}{32} ( \frac{109}{2} + \frac{17}{2} \sqrt{41})β)  −1
−100 = −4 · 52  10 + 5ω  52 · 41  2  ( \frac{9}{2} + \frac{7}{4} ω, − \frac{43}{16} − \frac{7}{16} \sqrt{41} + ( \frac{3}{8} \sqrt{41} + \frac{19}{8} )β)  −2
−115 = −115 · 1  177 + 68ω  41  2  (− \frac{31}{9} − \frac{11}{9} ω, \frac{73}{36} + \frac{11}{36} \sqrt{41} + \frac{1}{108} ( \frac{59}{5} + \frac{9}{5} \sqrt{41})β)  −1
−160 = −40 · 22  4ω  22 · 41  4  (32 + 12ω, −19 − 3\sqrt{41} + ( \frac{173}{2} + \frac{27}{2} \sqrt{41})β)  −2
−368 = −23 · 42  43 + 16ω  42 · 41  6  ( \frac{29}{4} + \frac{11}{4} ω, − \frac{69}{16} − \frac{11}{16} \sqrt{41} + ( \frac{13}{4} + \frac{1}{2} \sqrt{41})β)  −2
Table 4: ATR extensions of ℚ(\sqrt{41}) and ATR points on E{′ \atop 41}

4.4. Proof of Proposition 4.1.

The aim of this section is proving Proposition 4.1, which was left unproved in §4.1 and asserts that the relative discriminant of L/K factors as d(L/K) = c · N, where c is a positive integer such that L ⊂ L_{c} (and similarly L′ ⊂ L′_{c}).

Recall our assumption on N = disc(F) to be odd, and thus squarefree. Here we shall assume for notational simplicity that K ≠ ℚ(\sqrt{−1}) and ℚ(\sqrt{−3}), so that O_{K}^{×} = \{±1\}; we leave to the reader the task of filling the details for the two excluded fields.

Let us recall first the following classical lemma ( [16, §38-39]; cf. also Daberkow^{7}) on Kummer extensions of local fields, which applies in particular to our quadratic extension L/K.

Lemma 4.3. Let k be a local field containing all p-th roots of unity for some prime p and let v_{k} : k^{×} → ℤ denote the valuation map of k, normalized so that v_{k}(k^{×}) = ℤ. Let K/k be a Kummer extension of degree p with discriminant 𝔡_{K/k}. Then K = k(\sqrt[p]{ϑ}) for some ϑ ∈ k such that v_{k}(ϑ) ∈ {0, 1}. Moreover,

(i) If v_{k}(ϑ) = 1, v_{k}(𝔡_{K/k}) = pv_{k}(p) + (p − 1). (ii) Assume v_{k}(ϑ) = 0. If v_{k}(p) = 0, then v_{k}(𝔡_{K/k}) = 0. Otherwise, write

𝔭_{k} for the unique maximal ideal in k. We have:

(a) If equation x^{p} ≡ ϑ(mod 𝔭_{k}^{pv_{k}(p)/(p−1)}) can be solved in k, then

v_{k}(𝔡_{K/k}) = 0.

(b) If not, v_{k}(𝔡_{K/k}) = pv_{k}(p) + (p − 1)(1 − η), where η = max_{ℓ}{0 ≤

ℓ < pv_{k}(p)/(p − 1) | x^{p} ≡ ϑ(mod 𝔭_{k}^{ℓ}) can be solved in O_{k}}.

32

We use the above result in order to deduce several lemmas which shall allow us to reduce the proof of Proposition 4.1 to the case in which L/K is unramified at dyadic primes.

Lemma 4.4. Let p ∤ disc(K) be a prime and put p^{∗} = 8 if p = 2, p^{∗} = p if p ≡ 1(mod 4) and p^{∗} = −p if p ≡ −1(mod 4). Then K(\sqrt{p^{∗}}) is contained in the ring class field H_{c} of K associated to the order O_{c} of conductor c = |p^{∗}|.

Proof. Suppose first that p is split in K and fix a prime 𝔭|p in K. Let

U = K_{𝔭}^{×} ⋂ K^{×} ∏_{v} O_{c,v}^{×},

where the intersection is computed by regarding K_{𝔭}^{×} as a subgroup of ∏_{v} K_{v}^{×} by means of the usual embedding x_{𝔭} ↦ (1, ..., 1, x_{𝔭}, 1, ..., 1).

Since the map K_{𝔭}^{×}/U → 𝕀_{K}/ (K^{×} ∏_{v} O_{c,v}^{×}) is injective by [26, p. 173, Prop. 5.2], it follows that U ⊂ K_{𝔭}^{×} ≃ ℚ_{p}^{×} corresponds to H_{c,𝔓}/K_{𝔭} by local class field theory for any prime 𝔓 of H_{c} above 𝔭.

Write c = p^{r} with r = 3 if p = 2, r = 1 if p is odd. Since 1 + p^{r}ℤ_{p} ⊆ U, 1 + p^{r−1}ℤ_{p} ⊄ U by [4, p. 197], an easy calculation shows that

U = \{\frac{α}{\overline{α}} | α ∈ V \} · (1 + 𝔭^{r}),

where V = {α ∈ K^{×} |ord_{v}(α) = 0 ∀v ≠ 𝔭}. Note that V = {±α_{0}^{n}, n ∈ ℤ} for some α_{0} ∈ K^{×} such that ord_{v}(α_{0}) = 0 for all v ≠ 𝔭 and ord_{𝔭}(α_{0}) = n_{0} ≥ 1 is minimal. With this notation we have

U = \{(\frac{α_{0}}{\overline{α}_{0}} )^{n}, n ∈ ℤ\} · (1 + 𝔭^{r}).  (33)

Suppose now that p remains inert in O_{K}. Arguing similarly as before we obtain that the open subgroup U ⊂ K_{p}^{×} corresponding to H_{p^{r},𝔓}/K_{p} by local class field theory is U = K_{p}^{×} ∩ (K^{×} ∏_{v} O_{p^{r},v}^{×}), i.e.

\begin{aligned}U & = \{α | α ∈ K^{×}, ord_{v}(α) = 0, ∀v ≠ p\} · (1 + p^{r}O_{K_{p}}) \\ & = \{(±α_{0})^{n} | n ∈ ℤ\} · (1 + p^{r}O_{K_{p}}), \end{aligned}  (34)

where α_{0} ∈ K^{×} is chosen such that ord_{v}(α_{0}) = 0 for all v ≠ p and ord_{p}(α_{0}) ≥ 1 is minimal. We can thus take α_{0} = p^{n_{0}} for some n_{0} ≥ 1.

Put K′ = K(\sqrt{p^{∗}}). Any prime 𝔭 in K above p ramifies in K′. Fix one such prime 𝔭 and put 𝔭 = ℘^{2} in K′ so that K′_{℘} = K_{𝔭}(\sqrt{p^{∗}}). By class field theory, in order to prove that K′ ⊂ H_{c} it is enough to show that U ⊂ Nm_{K′_{𝔭}/K_{𝔭}}(K_{𝔭}^{′×}). Since d(K′/K) = p^{∗} by Lemma 4.3, K′ is contained in the ray class field K_{c} of conductor c of K and it thus suffices to verify

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that α_{0}\overline{α}_{0} (resp. ±α_{0}) lies in Nm_{K′_{𝔭}/K_{𝔭}}(K_{𝔭}^{′×}) if p splits (resp. remains inert) in K.

Assume p = 2. Then ±1, ±2 ∈ Nm_{K′_{℘}/K_{𝔭}}(K_{℘}^{′×}) because −1 = Nm(1 + \sqrt{2}) and −2 = Nm(\sqrt{2}). The lemma thus follows automatically if 2 is inert in K, while if 2 splits, it follows because α_{0}\overline{α}_{0} is a power of 2, hence α_{0}\overline{α}_{0} lies in either ±ℚ_{2}^{×^{2}} or ±2ℚ_{2}^{×^{2}}.

Assume p is odd. Then −p^{∗} = Nm(\sqrt{p^{∗}}) ∈ Nm_{K′_{𝔭}/K_{𝔭}}(K_{𝔭}^{′×}). Suppose first p splits in K: as before, it is enough to show that p ∈ Nm_{K′_{𝔭}/K_{𝔭}}(K^{′×}), which we already did if p^{∗} = −p. That the same holds when p^{∗} = p follows because p ≡ 1(mod 4) implies that −1 ∈ Nm_{K′_{𝔭}/K_{𝔭}}(K_{𝔭}^{′×}). Suppose now p remains inert in K; we must show that ±p ∈ Nm_{K′_{𝔭}/K_{𝔭}}(K_{𝔭}^{′×}). If p^{∗} = p this follows by the same reason as above; if p^{∗} = −p, then p ≡ 3 (mod 4), K_{p} = ℚ_{p}(\sqrt{−1}) and thus −1 ∈ K_{p}^{×^{2}}, which allows us to conclude.

Note that a direct consequence of the previous lemma is that for any odd square free integer m relatively coprime with disc(K) either K(\sqrt{m}) or K(\sqrt{−m}) is contained in H_{m}.

Lemma 4.5. d(L/K) = 2^{t}c_{0}N for some integer 0 ≤ t ≤ 3 and some positive integer c_{0} ≥ 1 relatively coprime to 2 and N. If further 2 is ramified in K, 0 ≤ t ≤ 2.

Proof.

Write K = ℚ(\sqrt{−d_{0}}) for some square free integer d_{0} > 0 and L = K(\sqrt{β}) for some β ∈ ℤ + ℤ\sqrt{−d_{0}} and square free in K. Without loss of generality, N can be written as N \overline{N} where N divides the square free part 𝔅 of (β) in K and \overline{N} is relatively coprime to 𝔅.

Write 𝔅_{2} for the largest ideal which divides 𝔅 and is relatively coprime to any prime of K above 2. Since v_{K_{𝔭′}} (2) = 0 and v_{K_{𝔭′}} (𝔅_{2}) = 1 for any prime 𝔭′ | 𝔅_{2}, Lemma 4.3 shows that v_{K_{𝔭′}} (𝔡_{L_{𝔓′} /K_{𝔭′}}) = 1, where 𝔓′ is the prime in L above 𝔭′, thus the prime-to-2 part of d(L/K) is 𝔅_{2}. Besides, 𝔅_{2} = N · ℭ with (ℭ, N) = 1. Since Nm_{K/ℚ}(β)/N is a perfect square in ℤ, ℭ is principal and can be written as ℭ = (c_{0}) for some integer c_{0} > 0. Hence N c_{0} | d(L/K) | 2^{t}N c_{0} for some integer t ≥ 0.

If 2 is unramified in K we have v_{K_{𝔭}}(2) = 1 for any prime 𝔭 | 2 in K and it follows from Lemma 4.3 that d(L/K) = N c_{0}2^{t} with 0 ≤ t ≤ 3.

Suppose now that 2 ramifies in K with (2) = 𝔭^{2} in K. Then, since v_{K_{𝔭}}(2) = 2 and Nm_{K/ℚ}(β)/N is a perfect square in ℤ, we fall into case (ii) of Lemma 4.3: for any prime 𝔓 in L above 𝔭, L_{𝔓} can be written as K_{𝔭}(\sqrt{ϑ})

34

for some ϑ ∈ K_{𝔭}^{×} such that v_{k_{𝔭}}(ϑ) = 0. Suppose v_{K_{𝔭}}(𝔡_{L_{𝔓}/K_{𝔭}}) ≠ 0. Then Lemma 4.3 (b) asserts that

v_{K_{𝔭}}(𝔡_{L_{𝔓}/K_{𝔭}}) = 5 − η,

where

η = max{0 ≤ ℓ < 4 | ∃ι ∈ O_{K_{𝔭}}, ι^{2} ≡ ϑ(mod 𝔭^{ℓ})}.

A classical result^{18}^{,}^{17} of Hilbert implies that v_{K_{𝔭}}(𝔡_{L_{𝔓}/K_{𝔭}}) is even. Hence d(L/K) = N 2^{t}c_{0} with 0 ≤ t ≤ 2.

Lemma 4.6. It is enough to prove Proposition 4.1 when d(L/K) = 2^{t}N and 0 ≤ t ≤ 2.

Proof. Lemma 4.5 shows in general d(L/K) = 2^{t}c_{0}N, where 0 ≤ t ≤ 3 and c_{0} ≥ 1. Suppose first that t = 3, then 2 is unramified in K by the same lemma. Let 𝔓 and 𝔭 be prime ideals in L and K respectively such that 𝔓|𝔭|2. Then L_{𝔓} can be written as K_{𝔭}(\sqrt{ϑ}) for some ϑ ∈ K with v_{K_{𝔭}}(ϑ) ∈ \{0, 1\}. Define L′ = K(\sqrt{ϑ′}), where ϑ′ is defined as

ϑ′ = \begin{cases}ϑ/2 & if v_{K_{𝔭}}(ϑ) = 1; \\ ϑ & if v_{K_{𝔭}}(ϑ) = 0. \end{cases}

Hence v_{K_{𝔭}}(ϑ′) = 0. Let 𝔓′ be a prime in L′ above 𝔭. Then either case (a) or (b) of Lemma 4.3 applies. In case (a), v_{K_{𝔭}}(𝔡_{L′_{𝔓′} /K_{𝔭}}) = 0. In case (b), v_{K_{𝔭}}(𝔡_{L′_{𝔓′} /K_{𝔭}}) = 3 − η, where 0 ≤ η ≤ 2, hence 𝔭 is ramified in L′, so residue field of L′_{𝔓′} is equal to that of K_{𝔭} and consequently η ≥ 1. We conclude that v_{K_{𝔭}}(𝔡_{L′_{𝔓′} /K_{𝔭}}) ≤ 2. By Lemma 4.4, L ⊂ K(\sqrt{2})L′ ⊂ H_{8}L′ with d(L′/K) = 2^{t′}c_{0}N for some integer 0 ≤ t′ ≤ 2.

Suppose now c_{0} > 1. Setting L′′ = K(\sqrt{δϑ′/c_{0}}) we have d(L′′/K) = 2^{t′}N, where δ ∈ \{±1\} such that K(\sqrt{δc_{0}}) ⊂ H_{c_{0}} as described in Lemma 4.4. By the same lemma, L′ ⊂ K(\sqrt{δc_{0}})L′′ ⊂ H_{c_{0}}L′′.

So L ⊂ H_{8}L′ ⊂ H_{8}H_{c_{0}}L′′ = H_{8c_{0}}L′′ such that L′′/K is a quadratic extension and d(L′′/K) = 2^{t′}N for some integer 0 ≤ t′ ≤ 2. This justifies the claim that we only need to prove proposition 4.1 when 0 ≤ t ≤ 2 and d(L/K) = 2^{t}N.

Thanks to Lemma 4.6 we can assume in what follows that c_{0} = 1 and 0 ≤ t ≤ 2.

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Lemma 4.7. There is a unique quadratic extension L_{2^{t}}/K_{2^{t}} contained in K_{2^{t}N} such that the set of primes in K_{2^{t}} which ramify in L_{2^{t}} is the set of primes above N. We have L ⊂ L_{2^{t}}.

Proof. Assume first t = 0 or 1. Then Gal(K_{2^{t}N} /K_{2^{t}}) ≅ (∏_{𝔭|N} (O_{K}/𝔭)^{×})/{±1} ≅ (ℤ/Nℤ)^{×}/{±1}. This is obvious for t = 0, and holds for t = 1 because K_{2}K_{N} = K_{2N}. Extension L_{2^{t}}/K_{2^{t}} corresponds by Galois theory to the unique primitive even quadratic Dirichlet character ε of conductor N.

Suppose now t = 2. Then

Gal(K_{2^{t}N} /K_{2^{t}}) ≅ G := ({±1} × (ℤ/Nℤ)^{×})/{±𝟙},

where 𝟙 = (1, 1) is the identity element of {±1} × (ℤ/Nℤ)^{×} . Again, any extension L_{2^{t}}/K_{2^{t}} as in the statement corresponds to a non-trivial character ε′ : G → {±1} which is trivial on {±1} × {1} and is even and primitive on {1} × (ℤ/Nℤ)^{×}. As above, the only such character is ε′ = 1 × ε.

Finally, note that LK_{2^{t}}/K_{2^{t}} is a quadratic extension contained in K_{2^{t}N}. Since disc(LK_{2^{t}}/K_{2^{t}}) = N it follows that L_{2^{t}} = LK_{2^{t}} and thus L ⊂ L_{2^{t}}.

Recall the quadratic extension L_{c} of the ring class field H_{c} introduced in §4.1, over which the Heegner points P_{c}^{+} and P_{c}^{−} ∈ CM(c) are rational. Lemma 4.7 reduces the proof of Proposition 4.1 to showing that L_{c} = L_{c}. Since L_{c} was defined as the quadratic extension of H_{c} cut out by the kernel of the single even primitive character ε of conductor N, it suffices to show that H_{2^{t}} = K_{2^{t}} for 0 ≤ t ≤ 2.

When t = 0 and we obviously have H_{1} = K_{1}. If t = 1 or 2, the ratio of the ray class number h_{2^{t}} by the ring class number h(O_{2^{t}}) is (cf. [26, p.154] for this and the remaining notations):

\frac{h_{2^{t}}}{h(O_{2^{t}})} = \frac{[U : U_{2^{t},1}]^{−1}Nm(2^{t}) ∏_{𝔭|2^{t}}(1 − \frac{1}{Nm(𝔭)})}{\frac{2^{t}}{[O_{K}^{×}:O_{2^{t}}^{×}]} ∏_{p|2^{t}}(1 − (\frac{d_{K}}{p} )\frac{1}{p})}

= \begin{cases}\frac{[O_{K}^{×}:O_{2^{t}}^{×}]}{[U:U_{2^{t},1}]} · \frac{2^{2t}(1− \frac{1}{4})}{2^{t}(1−(\frac{d_{K}}{2} )\frac{1}{2})} & if 2 is inert in K, \\ \frac{[O_{K}^{×}:O_{2^{t}}^{×}]}{[U:U_{2^{t},1}]} · \frac{2^{2t}(1− \frac{1}{2})^{2}}{2^{t}(1−(\frac{d_{K}}{2} )\frac{1}{2})} & if 2 is split in K, \\ \frac{[O_{K}^{×}:O_{2^{t}}^{×}]}{[U:U_{2^{t},1}]} · ^{\frac{2^{2t}(1− \frac{1}{2})}{2^{t}}} & if 2 is ramified in K. \end{cases}

= \frac{[O_{K}^{×} : O_{2^{t}}^{×}]}{[U : U_{2^{t},1}]} · 2^{t−1}.

(35)

36

Since K ≠ ℚ(\sqrt{−1}) and ℚ(\sqrt{−3}), [O_{K}^{×} : O_{2^{t}}^{×}] = 1. If t = 1, then U = U_{2^{t},1}, so K_{2} = H_{2}. If t = 2, then [U : U_{2^{t},1}] = 2, and therefore K_{2^{2}} = H_{2^{2}}.

References

1. C. Breuil, B. Conrad, F. Diamond, and R. Taylor. On the modularity of elliptic curves over Q: wild 3-adic exercises. J. Amer. Math. Soc. 14 (2001), no. 4, 843–939.

2. M. Bertolini and H. Darmon. Iwasawa’s main conjecture for elliptic curves over anticyclotomic ℤ_{p}-extensions. Ann. of Math. (2) 162 (2005), no. 1, 1–64.

3. D. Bump, S. Friedberg, and J. Hoffstein. Nonvanishing theorems for Lfunctions of modular forms and their derivatives. Invent. Math. 102 (1990), no. 3, 543–618.

4. D. A. Cox. Primes of the form x^{2} + ny^{2}: Fermat, class field theory, and complex multiplication. John Wiley & Sons (1989).

5. S. Comalada, Elliptic curves with trivial conductor over quadratic fields, Pacific J. Math. 144 (1990), 237-258.

6. J. E. Cremona. Modular symbols for Γ_{1}(N) and elliptic curves with everywhere good reduction. Math. Proc. Camb. Phil. Soc. (1992), 111, 199–218.

7. M. Daberkow. On computations in Kummer extensions. J. Symbolic Computation 31 (2001), 113–131.

8. H. Darmon and A. Logan. Periods of Hilbert modular forms and rational points on elliptic curves. International Math. Res. Not. 40 (2003), 2153–2180.

9. J. S. Ellenberg and C. Skinner. On the modularity of ℚ-curves. Duke Math. J. 109 (2001), no. 1, 97–122.

10. K. Fujiwara. Deformation rings and Hecke algebras in the totally real case. Preprint 1996, revised 2004, 2006.

11. J. Gärtner. Points de Darmon et variétés de Shimura. Thèse de Doctorat, Université Paris 7 (Jussieu) (2010).

12. S. Gelbart. Automorphic forms on adèle groups. Ann. of Math. Studies 83, Princeton Univ. Press, Princeton, NJ (1975).

13. E. González and X. Guitart. On the modularity level of modular abelian varieties over number fields. J. Number Theory 130 (2010), no. 7, 1560-1570.

14. X. Guitart and J. Quer. Modular abelian varieties over number fields. Preprint available at http://arxiv.org/abs/0905.2550.

15. B.H. Gross and D.B. Zagier. Heegner points and derivatives of L-series. Invent. Math. 84 (1986), no. 2, 225–320.

16. E. Hecke. Lectures on the theory of algebraic numbers. Translated from the German by George U. Brauer, Jay R. Goldman and R. Kotzen. Graduate Texts in Mathematics, 77. Springer-Verlag, New York-Berlin (1981).

17. D. Hilbert. Über die Theories des relativquadratischen Zahlkörpers. Math. Ann. 51 (1899), 1–127.

18. J.G. Huard, B.K. Spearman, and K.S. Williams. Integral bases for quartic fields with quadratic subfields. J. Number Theory 51 (1995), no. 1, 87–102.

19. F. Jarvis and J. Manoharmayum. On the modularity of supersingular elliptic curves over certain totally real number fields. J. Number Theory 128 (2008),

37

no. 3, 589–618.

20. V.A. Kolyvagin. Finiteness of E(Q) and LLI(E, Q) for a subclass of Weil curves. Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 3, 670–671; translation in Math. USSR-Izv. 32 (1989), no. 3, 523–541.

21. V. A. Kolyvagin and D. Yu. Logachev. Finiteness of the Shafarevich-Tate group and the group of rational points for some modular abelian varieties. Leningrad Math. J., 1 (1990), no. 5, 1229–1253.

22. V. A. Kolyvagin and D. Yu. Logachev. Finiteness of LLI over totally real fields. Math. USSR Izvestiya 39 (1992), no. 1, 829–853.

23. C. Khare and J.-P. Wintenberger. Serre’s modularity conjecture I, II. Invent. Math. 178 (2009), no. 3, 485–504 and 505–586.

24. M. Longo. On the Birch and Swinnerton-Dyer conjecture for modular elliptic curves over totally real fields. Ann. Inst. Fourier (Grenoble) 56 (2006), no. 3, 689–733.

25. J. S. Milne. On the arithmetic of abelian varieties. Invent. Math. 17 (1972), 177–190.

26. J. S. Milne. Class field theory. Available at http://www.jmilne.org/math.

27. M.R. Murty and V.K. Murty. Mean values of derivatives of modular L-series. Ann. of Math. (2) 133 (1991), no. 3, 447–475.

28. T. Oda. Periods of Hilbert modular surfaces. Progress in Mathematics, 19. Birkhäuser, Boston, Mass. (1982).

29. PARI/GP, version 2.3.4, Bordeaux (2008). Available at http://pari.math.ubordeaux.fr/.

30. A. Pacetti, On the change of root number under twisting and applications, available at arXiv:1010.3781.

31. R.G.E. Pinch. Elliptic curves over number fields. D. Phil thesis, Oxford University (1982).

32. K. A. Ribet. Abelian varieties over ℚ and modular forms, in Modular curves and abelian varieties. J. Cremona, J.-C. Lario, J. Quer, K. Ribet (eds.), Progress in Mathematics 224, Birkhäuser (2004), 241–261.

33. D.E. Rohrlich. Nonvanishing of L-functions and structure of Mordell-Weil groups. Journal für di Reine und Angewand. Math. 417 (1991), 1–26.

34. D.E. Rohrlich. Variation of the root number in families of elliptic curves. Compositio Math. 87 (1993) 119–151.

35. G. Shimura. Introduction to the arithmetic theory of automorphic functions. Reprint of the 1971 original. Publications of the Mathematical Society of Japan, 11. Kanô Memorial Lectures, 1. Princeton Univ. Press, Princeton, NJ (1994).

36. K. Shiota. On the explicit models of Shimura’s elliptic curves. J. Math. Soc. Japan 38 (1986), no. 4, 649–659.

37. C. Skinner and A. Wiles. Nearly ordinary deformations of irreducible residual representations. Ann. Fac. Sci. Toulouse Math. (6) 10 (2001), 185–215.

38. R. Taylor and A. Wiles. Ring-theoretic properties of certain Hecke algebras. Ann. of Math. (2) 141 (1995), no. 3, 553–572.

39. Y. Tian and S. Zhang. Book, in progress.

40. J.-L. Waldspurger. Sur les valeurs de certaines fonctions L automorphes en

38

leur centre de symmétrie. Compositio Math. 54 (1985), 173–242.

41. A. Wiles. Modular elliptic curves and Fermat’s last theorem. Ann. of Math. (2) 141 (1995), no. 3, 443–551.

42. X. Yuan, W. Zhang, and S. Zhang. Heights of CM points I: Gross–Zagier formula. Preprint.

43. S. Zhang. Heights of Heegner points on Shimura curves. Ann. of Math. (2) 153 (2001), no. 1, 27–147.

44. S. Zhang. Arithmetic of Shimura curves. Science China Mathematics 53 (2010), 573–592.

45. Y. Zhao. McGill Univ. Ph. D Thesis. In progress.