CLASS NUMBERS OF QUADRATIC FIELDS AND SHIMURA’S CORRESPONDENCE

Jan Nekovář

Mathematical Institute of the Czechoslovak Academy of Science

I. Introduction

It is well-known (see Fueter [8] and Aigner [1]) that Fermat’s equation X^{3}+Y^{3} = 1 has no solutions in a quadratic field K = Q(\sqrt{D}) ( for |D| ≡ 1 (mod 3)), provided the class number of K is not divisible by 3. This subject was further investigated by Frey [6], who gave estimates for the 3-rank of the class group of K in terms of 3-descent on a curve Y^{2} = X^{3} + D.

In this paper we consider curves E_{D} : DY^{2} = 4X^{3} − 27 ,which are quadratic twists of the Fermat’s curve X^{3} +Y^{3} = 1. We give a precise formula for the rank of the Selmer group corresponding to the complex multiplication \sqrt{−3} : E_{D} ⟶ E_{−3D} in terms of the 3-rank of the class group of Q(\sqrt{D}) resp. Q(\sqrt{−3D}). This result may be considered as a quantitative version of [1] and [8].

We discuss also Birch and Swinnerton-Dyer’s conjecture for curves E_{D}. According to a theorem of Waldspurger [22], [23], natural rational factor of L(E_{D}/Q, 1) may be expressed in terms of coefficients of certain modular forms of weight 3/2. We identify these forms explicitely and verify (mod 3)-part of Birch and Swinnerton-Dyer’s conjecture for E_{D} in most cases (for D of density 5/6 ).

Typeset by AMS-TEX

II. Descent on curves E_{D}

1. Notation and preliminaries.

Let D ≠ 1 be a square-free integer prime to 3.Put D_{+} = max(D, −3D), D_{−} = min(D, −3D), K_{±} = Q(\sqrt{D_{±}}), K_{−3} = Q(\sqrt{−3}), L = K_{+}K_{−}. Let G(L/Q) = {1, χ_{+}, χ_{−}, χ_{−3}} be the character group of the Galois group G(L/Q), where Ker (χ_{±}) = G(L/K_{±}), Ker(χ_{−3}) = G(L/K_{−3}). Denote by C, C_{±} the class groups of L, K_{±} respectively and by _{p}_{∞}C,_{p}_{∞}(C_{±}) their p-primary parts. For an odd prime p we identify _{p}_{∞}(C_{±}) with their images in C under canonical morphisms C_{±} ⟶ C.

We consider the Fermat curve E : x^{3} + y^{3} = 1, which is isomorphic (over Q) to Y^{2} = 4X^{3} − 27, an isomorphism being given by the following formulas:

X = \frac{3}{x + y} , Y = \frac{9(x − y)}{x + y}, x = \frac{9 + Y}{6X} , y = \frac{9 − Y}{6X} .

Endomorphism ring of E is End_{K_{−3}}(E) = Z[ρ], where ρ = \frac{−1+\sqrt{−3}}{2} acts by

ρ(x, y) = (ρ^{−1}x, ρ^{−1}y)  and  ρ(X, Y) = (ρX, Y)

in Fermat’s and Weierstrass’ coordinates respectively.

We shall also consider twisted forms

E_{±} : D_{±}Y^{2} = 4X^{3} − 27, E_{−3} : Y^{2} = 4X^{3} + 1

of E. Let λ = ρ−ρ^{2} = \sqrt{−3} ∈ End(E) be the unique (up to a root of unity) isogeny of degree 3. It is defined over K_{−3}, but induces isogenies E ⟷ E_{−3}, E_{+} ⟷ E_{−} defined already over Q (see §2).

If G is a finite abelian group, χ ∈ G a character of G with values in a ring A and M an A[G]-module, put

M_{χ} = {m ∈ M | gm = χ(g)m for every g ∈ G}.

We shall use the notation H^{i}(K/k, A) resp. H^{i}(k, A) for the Galois cohomology groups H^{i}(G(K/k), A) resp. H^{i}(G(k̄/k), A).

CLASS NUMBERS OF QUADRATIC FIELDS AND SHIMURA’S CORRESPONDENCE 3

2. Twisted isogenies.

In this section we recall some basic facts about twisted forms of abelian varieties and isogenies. We are interested only in the case of the Fermat cubic, but we prefer to give the statements first in the general context.

Let A be an abelian variety over a field K, L/K a finite Galois extension. Isomorphism classes of abelian varieties A′ over K such that A′ ×_{K} L ≃ A ×_{K} L are in a bijective correspondence with elements of the pointed set H^{1}(L/K, Aut_{L}(A)): every isomorphism

f : A ×_{K} L ⟶^{∼} A′ ×_{K} L defines a 1-cocycle
a(g) = ^{g}f^{−1} ∘ f ∈ Aut_{L}(A) (g ∈ G(L/K)).

Cohomology class of a(g) depends only on the isomorphism class of A′. Denote by A(a) the abelian variety A′ corresponding to the cocycle a(g).

Suppose we are given a separable isogeny λ : A ×_{K} L ⟶ A ×_{K} L and

isomorphisms f : A ×_{K} L ⟶^{∼} A′ ×_{K} L, f′ : A ×_{K} L ⟶^{∼} A′′ ×_{K} L such that

the isogeny λ′ = f′ ∘ λ ∘ f^{−1} : A′ ×_{K} L ⟶ A′′ ×_{K} L is already defined over K. Then the following diagram is commutative for every g ∈ G(L/K):

(2.1)  \begin{matrix}A′ ×_{K} L & ⟵^{f} & A ×_{K} L & ⟶^{λ} & A ×_{K} L & ⟶^{f′} & A′′ ×_{K} L \\ ∥ & & ↓^{a(g)} & & ↓^{a′(g)} & & ∥ \\ A′ ×_{K} L & ⟵^{^{g}f} & A ×_{K} L & ⟶^{^{g}λ} & A ×_{K} L & ⟶^{^{g}f′} & A′′ ×_{K} L \end{matrix},

where a(g), a′(g) are cocycles corresponding to f and f′. We say that λ is \underline{L}/\underline{K} − \underline{admissible} if Ker(λ) is G(L/K)−invariant. If this is the case, it follows from the diagram (2.1) that

a(g) ∈ Aut_{L}(A, λ) := {a ∈ Aut_{L}(A) | a(Ker(λ)) = Ker(λ) }. Conversely, suppose that λ is L/K−admissible and a(g) is a 1-cocycle with val-

ues in Aut_{L}(A, λ). Such data determine uniquelly automorphisms a′(g) ∈ Aut_{L}(A) making (2.1) commutative. These a′(g) form a 1-cocycle, hence define a twisted

form f′ : A ×_{K} L ⟶^{∼} A′′ ×_{K} L of A such that λ_{a} := f′ ∘ λ ∘ f^{−1} :

A(a) ⟶ A(a′) is defined over K. Cocycles cohomologous to a(g) give isogenies K−isomorphic to λ_{a}. If End_{L}(A) is commutative, then Aut_{L}(A, λ) = Aut_{L}(A) and a′(g) = c(g)a(g), where c(g) ∈ Aut_{L}(A) satisfies ^{g}λ = c(g)λ.

We summarize previous discussion in the following

4 JAN NEKOVÁŘ

Proposition 2.1. Let λ : A ×_{K} L ⟶ A ×_{K} L be an L/K−admissible separable isogeny. Then we have a commutative diagram, whose horizontal arrows are bijections:

\{ \begin{matrix}isomorphism classes (over K) \\ of isogenies λ′ : A′ ⟶ A′′ \\ with λ′ ×_{K} L ≃ λ \end{matrix} \} ⟶^{1−1} H^{1}(L/K, Aut_{L}(A, λ))
↓^{forget λ′,A′′}  ↓^{canonical}
\{ \begin{matrix}isomorphism classes (over K) \\ of A′ with A ×_{K} L ≃ A′ ×_{K} L \end{matrix} \} ⟶^{1−1} H^{1}(L/K, Aut_{L}(A))

Cocycles a(g), a′(g) representing twisted forms A′, A′′ satisfy

gλ ∘ a(g) = a′(g) ∘ λ.

If End_{L}(A) is commutative, then a′(g) = c(g)a(g), where

gλ = c(g)λ. □

We apply Proposition 2.1 in the following situation: K = Q, L = Q(\sqrt{D}, \sqrt{−3D}), A = E : x^{3} + y^{3} = 1, λ = ρ − ρ^{2} = \sqrt{−3}. As ^{g}λ = χ_{−3}(g)λ (g ∈ G(L/Q)), λ is L/K−admissible. We have

H^{1}(L/Q, Aut_{L}(E)) = H^{1}(L/Q, μ_{6}) = H^{1}(L/Q, μ_{2}) = G(L/Q) ,

so the L/Q−forms of E are the following curves:

E_{−3} = E(χ_{−3}) : Y^{2} = 4X^{3} + 1, E : Y^{2} = 4X^{3} − 27
E_{±} = E(χ_{±}) : D_{±}Y^{2} = 4X^{3} − 27,

corresponding to the characters 1, χ_{±}, χ_{−3} respectively. According to Proposition 2.1, λ induces isogenies λ_{χ} : E(χ) ⟶ E(χχ_{−3}) defined over Q for every χ ∈ G(L/Q) .

Explicit formulas for λ_{χ} are easily obtained from a more general isogeny between curves y^{2} = 4x^{3} + a and y^{′2} = 4x^{′3} − 27a:

x′ = \frac{x^{3} + a}{x^{2}}, y′ = \frac{y(x^{3} − 2a)}{x^{3}}.

3. Descent for twisted isogenies.

CLASS NUMBERS OF QUADRATIC FIELDS AND SHIMURA’S CORRESPONDENCE 5

Suppose K is a number field, A, A′ are abelian varieties over K and λ : A ⟶ A′ an isogeny also defined over K. The exact sequence

0 ⟶ Ker(λ) ⟶ A(K̄) ⟶^{λ} A′(K̄) ⟶ 0

induces a commutative diagram

0 ⟶ A′(K)/λA(K) ⟶^{α} H^{1}(K, Ker(λ))
↓ ↓^{β_{v}}
0 ⟶ A′(K_{v})/λA(K_{v}) ⟶^{α_{v}} H^{1}(K_{v}, Ker(λ))

for every prime divisor v of K and the corresponding completion K_{v} (including archimedean v). Selmer group of λ is defined as

S(λ, A/K) = ⋂_{v} β_{v}^{−1}(Im(α_{v})).

It sits in an exact sequence

0 ⟶ A′(K)/λA(K) ⟶ S(λ, A/K) ⟶ _{λ}\underline{III}(A/K) ⟶ 0,

where _{λ}\underline{III}(A/K) is the subgroup of λ−torsion elements of the Tate-S̆afarevic̆ group of A over K.

For A = A′ = E : x^{3} + y^{3} = 1, λ = \sqrt{−3} one has Ker(λ) = μ_{3}, H^{1}(K, Ker(λ)) ≃ K^{∗}/K^{∗3}. Suppose μ_{3} ⊂ K. Then a ∈ K^{∗}/K^{∗3} lies in Im(α) iff the curve D_{a} : a^{−1}x^{3} + ay^{3} = 1 contains a K-ratinal point and a ∈ S(λ, E/K) iff D_{a} contains a K_{v}-rational point for all v (see [3]).

Suppose that in the situation of Proposition 2.1, L/K is a Galois extension of number fields. We shall compare Selmer groups of λ and λ′. Proposition 3.1. Let L/K be a finite Galois extension, A an abelian variety over K, λ : A ×_{K} L ⟶ A ×_{K} L a separable L/K-admissible isogeny of degree prime to [L : K]. Let a(g), a′(g) be a pair of 1-cocycles corresponding to a K-isogeny λ_{a} : A′ = A(a) ⟶ A′ = A(a′) as in Proposition 2.1. Then we have a commutative diagram whose vertical arrows are all isomorphisms:

0 ⟶  A′′(K)/λ_{a}A′(K)  ⟶ H^{1}(K, Ker(λ_{a}))
↓  ↓^{res}
0 ⟶ (A′′(L)/λ_{a}A′(L))^{G(L/K)} ⟶ H^{1}(L, Ker(λ_{a}))^{G(L/K)}
↑^{f′_{∗}}  ↑^{f_{∗}}
0 ⟶  (A(L)/λA(L))_{a′}  ⟶ H^{1}(L, Ker(λ))_{a}

6 JAN NEKOVÁŘ

Here M_{a} = {m ∈ M | ^{g}m = a(g)m ∀g ∈ G(L/K)}.

Proof . Commutativity of the upper square is obvious. As deg(λ) is prime to [L : K], the Hochschild-Serre spectral sequence degenerates into isomorphisms

res : H^{i}(K, Ker(λ_{a})) ⟶^{∼} H^{i}(L, Ker(λ_{a}))^{G(L/K)}.

From exact sequences

0 ⟶ (Ker(λ_{a}))(K) ⟶ A′(K) ⟶ (A′(L)/(Ker(λ_{a}))(L))^{G(L/K)} ⟶ 0
0 ⟶ (A′(L)/(Ker(λ_{a}))(L))^{G(L/K)} ⟶ A′′(K) ⟶
⟶ (A′′(L)/λ_{a}A′(L))^{G(L/K)} ⟶ 0

it follows that the upper left vertical arrow is also an isomorphism.

From the diagram (2.1) we get exact sequences

0 ⟶ Ker(λ) ⟶ A(K̄) ⟶^{λ} A(K̄) ⟶ 0
↓^{f} ↓^{f} ↓^{f′}
0 ⟶ Ker(λ_{a}) ⟶ A′(K̄) ⟶^{λ_{a}} A′′(K̄) ⟶ 0

(the vertical arrows are isomorphisms of G(K̄/L)−modules, but not of G(K̄/K)- modules). By taking cohomology over L, we get a commutative diagram

\begin{aligned}0 & ⟶ & A(L)/λA(L) & ⟶ & H^{1}(L, Ker(λ)) \\ & & ↓^{f′} & & ↓^{f_{∗}} \\ 0 & ⟶ & A′′(L)/λ_{a}A′(L) & ⟶ & H^{1}(L, Ker(λ_{a})) \end{aligned}.

Suppose that F (h)(h ∈ G(K̄/L)) is a 1-cocycle representing a cohomology class [F] ∈ H^{1}(L, Ker(λ)). Then f_{∗}[F] is represented by f_{∗}F = f ∘ F. But

f_{∗}F = (^{g}f)(a(g)F (h))  and
(^{g}(f_{∗}F))(h) = ^{g}(f(F (g^{−1}hg))) = (^{g}f)((^{g}F)(h))

for g ∈ G(L/K), hence

(^{g}(f_{∗}F) − f_{∗}F)(h) = (^{g}f)((^{g}F)(h) − a(g)F (h))

and ^{g}(f_{∗}F) − f_{∗}F is a coboundary iff ^{g}F − a(g)F is. Similarly, for x ∈ A(L)/λA(L) and g ∈ G(L/K),

g(f′(x)) − f′(x) = (^{g}f′)(^{g}x − a′(g)x).

This proves that both f_{∗} and f′_{∗} are isomorphisms. □

CLASS NUMBERS OF QUADRATIC FIELDS AND SHIMURA’S CORRESPONDENCE 7

Proposition 3.2. Under the hypotheses of Proposition 3.1, suppose that K is a number field. Then the map

f_{∗}^{−1} ∘ res : H^{1}(K, Ker(λ_{a})) ⟶^{∼} H^{1}(L, Ker(λ))_{a}

induces an isomorphism

S(λ_{a}, A/K) ⟶^{∼} S(λ, A ×_{K} L/L)_{a}.

Proof . Consider the commutative diagram of Proposition 3.1

A′′(K)/λ_{a}A′(K) ⟶ H^{1}(K, Ker(λ_{a}))
↓^{f_{∗}^{′−1}} ↓^{f_{∗}^{−1}∘res}
(A(L)/λA(L))_{a′} ⟶ H^{1}(L, Ker(λ))_{a}

and the analogous diagrams for all completions K_{v}. The statement of the Proposition follows by a trivial diagram chase. □ Corollary 3.3. For A = E : x^{3} + y^{3} = 1, λ = \sqrt{−3}, L/K = Q(\sqrt{D}, \sqrt{−3D})/Q and χ ∈ G(L/K) we have

S(λ, E/L)_{χ} = S(λ_{χ}, E_{χ}/Q)
S(λ, E/L) = ⨁ ; χ∈G(L/Q) S(λ_{χ}, E_{χ}/Q).

4. Explicit calculation of Selmer groups.

In this section we show how the Selmer groups of isogenies λ_{χ} are related to the class groups of K_{±}.

Let H be the Hilbert class field of L. Artin’s reciprocity law yields an isomorphism of G(L/K)-modules

ψ : C ≃ G(H/L)

(G(L/Q) acts on G(H/L) by inner automorphisms). Let F be the subextension of H/L corresponding to the subgroup C^{3}. Again

ψ : C/C^{3} ≃ G(F/L)

is an isomorphism of G(L/Q)-modules. By Kummer’s theory we get a monomorphism

f : Hom(C/C^{3}, μ_{3}) ≃ Hom(G(F/L), μ_{3}) ↪ H^{1}(L, μ_{3}) = L^{∗}/L^{∗3}.

In other words, F = L(\sqrt[3]{a_{1}}, ⋯, \sqrt[3]{a_{r}}) with a_{i} ∈ L^{∗} and Im(f) is generated by the images of a_{i}′s in L^{∗}/L^{∗3}.

8 JAN NEKOVÁŘ

Proposition 4.1. The image of

f : Hom(C/C^{3}, μ_{3}) ⟶ H^{1}(L, μ_{3}) = H^{1}(L, Ker(λ))

is contained in the Selmer group S(λ, E/L).

Proof . According to the preceding discussion, a ∈ L^{∗}/L^{∗3} lies in Im(f) iff L(\sqrt[3]{a})/L is unramified, i.e. iff for all (non-archimedean) prime divisors v of L the image of a under β_{v} : H^{1}(L, Ker(λ)) ⟶ H^{1}(L_{v}, Ker(λ)) is contained in the group of unramified cohomology classes H_{ur}^{1}(L_{v}, Ker(λ)). As deg(λ) = 3 and E has good reduction outside 3,

H_{ur}^{1}(L_{v}, Ker(λ)) = Im[α_{v} : E(L_{v})/λE(L_{v}) ⟶ H^{1}(L_{v}, Ker(λ))]

for each v of residue characteristic different from 3 (see [5]).

We are thus reduced to prove that if L(\sqrt[3]{a})/L) is unramified and v is a prime divisor of L dividing 3, then the curve D_{a} : a^{−1}x^{3}+ay^{3} = 1 contains an L_{v}-rational point.

Let O_{v} be the ring of integers in L_{v}. Its prime element is π = ρ − ρ^{2} and the residue field O_{v}/πO_{v} has q = 3 or 9 elements. Put U_{n} = 1 + π^{n}O_{v}. Then L_{v}^{∗} = π^{Z} ×μ_{q−1} ×U_{1} . We may assume, therefore, that a ∈ U_{1}. Let x^{3} = a, M = L_{v}(x). As M/L_{v} is unramified and x^{3} ≡ 1 (mod π), one must also have x ≡ 1 (mod π), which implies that a = x^{3} ≡ 1 (mod π^{3}). Then −ρa − ρ^{−1}a^{−1} ≡ 1 (mod π^{4}) is a cube in L_{v}, as U_{4} = U_{2}^{3}. This means that D_{ρa} contains an L_{v}-rational point, so ρa ∈ Im(α_{v}). As D_{ρ} contains the point (-1,-1), a ∈ Im(α_{v}) as well. □ Corollary 4.2. f induces monomorphisms

f_{±} : Hom(C_{∓}/C_{∓}^{3}, μ_{3}) ↪ S(λ_{±}, E_{±}/Q).

Proof . As μ_{3} = (μ_{3})_{χ_{−3}}, our claim follows from Corollary 3.3 and the following Lemma 4.3. If p > 2 is a prime, then

(_{p}_{∞}C)_{χ_{±}} = _{p}_{∞}(C_{±}), (_{p}_{∞}C)_{1} = (_{p}_{∞}C)_{χ_{−3}} = 0.

Proof of the Lemma. As _{p}_{∞}(C_{±}) = (_{p}_{∞}(C_{±}))_{χ_{±}}, the morphism

p_{∞}(C_{+}) ⊕_{p}_{∞}(C_{−}) ⟶^{+} _{p}_{∞}C

CLASS NUMBERS OF QUADRATIC FIELDS AND SHIMURA’S CORRESPONDENCE 9

is injective. On the other hand, the class number formulas together with the identity

ζ^{2}(s)ζ_{L}(s) = ζ_{K_{+}}(s)ζ_{K_{−}}(s)ζ_{K_{−3}}(s)

imply that # (_{p}_{∞}C) = # (_{p}_{∞}(C_{+})) # (_{p}_{∞}(C_{−})). □

Before we proceed further, we need some information about bad fibres of minimal models of E_{±}. We recall that D is a square-free integer prime to 3 and we allow for the moment also D = 1. We change coordinates on E_{±} as follows : let

E : Y^{2} = X^{3} − 2^{4} · 3^{3} · D^{3}, E′ : Y^{′2} = X^{′3} + 2^{4} · D^{3}

and let ω = \frac{dX}{2Y} , ω′ = \frac{dX′}{2Y′} be regular differentials on E resp. E′. Proposition 4.4. For a prime p, let c_{p} resp. c′_{p} denote the number of components of the fibre over p of the Néron model of E resp. E′. Let ω_{min} resp. ω′_{min} be a Néron differential of E resp. E′ over Z. Then

(1) The numbers of components of Néron models are

\begin{aligned}c_{p} & = c′_{p} = 1 \\ c_{p} & = c′_{p} = 1 + #\{x ∈ F_{p} | x^{3} = 2\} \\ c_{3} & = 1 + #\{x ∈ F_{3} | x^{2} = D\}, c′_{3} = 1 \\ c_{2} & = c′_{2} = 1 \end{aligned}  \begin{matrix}for p |̸ 6D \\ for p|D, p ≠ 2 \end{matrix}

The common conductor of both E, E′ is

N = 3^{3}D^{2} × \{ \begin{matrix}1, & if D ≡ 1 (mod 4) \\ 2^{4}, & if D ≡ 2, 3 (mod 4) \end{matrix}

(2) The minimal equations of E, E′ over Z are

\begin{aligned}Y^{2} & = X^{3} − 2^{4} · 3^{3} · D^{3} \\ Y^{2} & = X^{3} − 2 · 3^{3} · (D/2)^{3} \\ Y^{2} + Y & = X^{3} − (1 + 27D^{3})/4 \end{aligned}  \begin{aligned}Y^{′2} & = X^{′3} + 2^{4} · D^{3} \\ Y^{′2} & = X^{′3} + 2 · (D/2)^{3} \\ Y^{′2} + Y′ & = X^{′3} + (D^{3} − 1)/4 \end{aligned}

for D ≡ 3, 2, 1 (mod 4) respectively.

\frac{ω_{min}}{ω} = \frac{ω′_{min}}{ω′} = \{ \begin{matrix}1, & for D ≡ 3 (mod 4) \\ 2, & for D ≡ 1, 2 (mod 4) \end{matrix}

(3)

∫_{E(R)} |ω|_{∞} = \frac{1}{2}Ω|D|^{−1/2} × \{ \begin{matrix}1, & for D < 0 \\ 3^{−1/2}, & for D > 0 \end{matrix}

∫_{E′(R)} |ω′|_{∞} = \frac{1}{2}Ω|D|^{−1/2} × \{ \begin{matrix}1, & for D < 0 \\ 3^{1/2}, & for D > 0 \end{matrix} ,

where Ω = \frac{1}{2π} Γ (\frac{1}{3})^{3} = 3, 059908 … is the real period of y^{2} = 4x^{3} − 1.

(4) The torsion subgroups over Q are

# (E(Q)_{tors}) = 1, # (E′(Q)_{tors}) = \{ \begin{matrix}3, & for D = 1 \\ 1, & for D ≠ 1 \end{matrix}

Proof . (1) and (2) follow from Tate’s algorithm [21].

(3) is an elementary calculation.

(4) As the reduction map is injective in the case of good reduction, # (E(Q)_{tors}) divides n := g.c.d.{E(F_{p}) | p |̸ 6D}. As all p ≡ 2 (mod 3) prime to 2D are supersingular for E, #E(F_{p}) = p + 1 for such p, hence n|3. It follows that

E(Q)_{tors} ⊆ Ker(λ) = \{0, (0, ±12D\sqrt{−3D})\}

and similarly E′(Q)_{tors} ⊆ Ker(λ′) = \{0, (0, ±4D\sqrt{D})\}. □

To get an upper bound for the Selmer groups in question, we note that, according to the Hilbert Theorem 90,

(L^{∗}/L^{∗3})_{χ_{±}} = {x ∈ K_{±}^{∗}/K_{±}^{∗3} | N_{K_{±}/Q}(x) ∈ Q^{∗3}}
(L^{∗}/L^{∗3})_{χ_{−3}} = {x ∈ K_{−3}^{∗}/K_{−3}^{∗3} | N_{K_{−3}/Q}(x) ∈ Q^{∗3}}
(L^{∗}/L^{∗3})_{1} = Q^{∗}/Q^{∗3}

All of the groups S(λ_{±}, E_{±}/Q), C_{±}/C_{±}^{3} are vector spaces over F_{3}. Denote by s_{±}, r_{±} respectively their dimensions over F_{3}. Then

#(C_{±}/C_{±}^{3}) = 3^{r_{±}}, #(C/C^{3}) = 3^{r_{+}+r_{−}}, #S(λ_{±}, E_{±}/Q) = 3^{s_{±}}.

Proposition 4.5.

(1) s_{+} ≤ 1 + r_{+}, s_{−} ≤ r_{−}.

(2) S(λ_{1}, E/Q) = 0, S(λ_{−3}, E_{−3}/Q) = Z/3Z.

(3) s_{+} − s_{−} = \{ \begin{matrix}0, & for |D| ≡ 1 (mod 3) \\ 1, & for |D| ≡ 2 (mod 3) \end{matrix}

Proof . (1) S(λ_{+}, E_{+}/Q) is the subgroup of all a ∈ K_{+}^{∗}/K_{+}^{∗3} such that N_{K_{+}/Q}(a) ∈ Q^{∗3} and the curve D_{a} : a^{−1}x^{3} + ay^{3} = 1 admits a rational point in all completions K_{+,v}. For any such a the principal divisor (a) = b^{3} must be a cube of some

CLASS NUMBERS OF QUADRATIC FIELDS AND SHIMURA’S CORRESPONDENCE11

divisor b. This defines a homomorphism φ(a) = b, φ : S(λ_{+}, E_{+}.Q) ⟶ _{3}C_{+} with Ker(φ) ⊆ W_{+}/W_{+}^{3} = Z/3Z, where W_{+} denotes the group of units of K_{+}. It follows that 3^{s_{+}} ≤ 3^{1+r_{+}}. One obtains similarly S(λ_{−}, E_{−}/Q) ↪ _{3}C_{−} and s_{−} ≤ r_{−}.

(2) The upper bounds #S(λ_{1}, E/Q) ≤ 1, #S(λ_{−3}, E_{−3}/Q)|3 are obtained in the same way as in (1). The torsion subgroup of E_{−3}(Q) is responsible for nontriviality of S(λ_{−3}, E_{−3}/Q).

(3) According to the duality theorem of Cassels [4],

\frac{#S(λ_{+}, E_{+}/Q)}{#S(λ_{−}, E_{−}/Q)} = (\frac{#E_{+}(Q)_{tors}}{#E_{−}(Q)_{tors}} )^{2} ∏_{p} \frac{c_{p}(E_{−})}{c_{p}(E_{+})} \frac{∫_{E_{−}(R)} |ω_{min}^{−}|_{∞}}{∫_{E_{+}(R)} |ω_{min}^{+}|_{∞}}

and the R.H.S. is calculated by using Proposition 4.4. □

Theorem 4.6. Let D ≠ 1 be a square-free integer prime to 3, D_{+} = max(D, −3D), D_{−} = min(D, −3D). Let r_{+} , r_{−} be the ranks of the 3primary parts of the class groups of Q(\sqrt{D_{+}}), Q(\sqrt{D_{−}}) and s_{+}, s_{−} the ranks of the Selmer groups of isogenies E_{+} ⟶^{λ_{+}} E_{−} resp. E_{−} ⟶^{λ_{−}} E_{+} of degree 3. Then

(1) r_{+} ≤ s_{−} ≤ r_{−} ≤ s_{+} ≤ 1 + r_{+}

(2) there exist natural exact sequences

0 ⟶ Hom(C_{∓}/C_{∓}^{3}, μ_{3}) ⟶ S(λ_{±}, E_{±}/Q) ⟶ A_{±} ⟶ 0

with

A_{+} = 0,_{A_{−} = 0,} #A_{−} = 3^{r_{−}−r_{+}},_{#A_{+} = 3^{1+r_{+}−r_{−}},} if |D| ≡ 1 (mod 3)

if |D| ≡ 2 (mod 3)

(3) The ranks of the Selmer groups are given by

s_{+} = s_{−} = r_{−},_{s_{+} = 1 + r_{+}, s_{−} = r_{+},} if |D| ≡ 1 (mod 3)

if |D| ≡ 2 (mod 3)

Proof . (1) By Corollary 4.2 r_{+} ≤ s_{−}, r_{−} ≤ s_{+} and by Proposition 4.5.1 s_{−} ≤ r_{−}, s_{+} ≤ 1 + r_{+}. (2),(3) Follow from (1), Corollary 4.2 and Proposition 4.5.3. □

12 JAN NEKOVÁŘ

Corollary 4.7. (Reichardt [13],Scholz [15]) r_{+} ≤ r_{−} ≤ 1 + r_{+}.

Corollary 4.8. (Fueter [8], Aigner [1]) If |D| ≡ 1 (mod 3) and r_{−} = 0, then E(K_{+}) = E(K_{−}) = E(Q) = Z/3Z.

Proof . If A is a curve A : y^{2} = x^{3}+ax+b, L/K = K(\sqrt{D}/K a quadratic extension with Galois group G = {1, c}, define two morphisms f, g : A(L) ⟶ A(L) by f(P) = P −^{c}P, g(P) = P +^{c}P. One has an exact sequence

0 ⟶ A(K) ⟶ A(L) ⟶^{f} Ker(g) ⟶ H^{1}(G, A(L)) ⟶ 0

A point P = (x, y) lies in Ker(g) iff (x, yD^{−1/2}) ∈ A_{D}(K), where A_{D} : Dy^{2} = x^{3} + ax + b. It follows that the sequence

0 ⟶ A(K) ⟶ A(L) ⟶ A_{D}(K) ⟶ H^{1}(G, A(L)) ⟶ 0

is exact. Take A = E, L/K = K_{±}/Q, in which case A_{D} = E_{±}. If |D| ≡ 1 (mod 3) and r_{−} = 0, then E_{±}(Q) = 0 by Theorem 4.6.3 and E(K_{±}) = E(Q) = Z/3Z by Proposition 4.4.4. □

III. Modular forms

5. L-function of E_{D}.

L-series of the curve E is given by the well-known formula (see e.g. [10, p.313])

L(E/Q, s) = ∑_{α≡1 (3)} α(Nα)^{−s},

where α runs through all elements of Z[ρ] congruent to 1 (mod 3). It is also classical [7, vol.II, p.388] that E is isomorphic to the (compactified) modular curve X_{0}(27) ; if

η(z) = q^{1/24} ∏_{n=1}^{∞}(1 − q^{n}), q = e^{2πiz}

is the Dedekind function and η_{a}(z) = η(az), then

X = \frac{η_{9}^{4}}{η_{3}η_{27}^{3}} , Y = 2 (\frac{η_{3}}{η_{27}} )^{3} + 9

CLASS NUMBERS OF QUADRATIC FIELDS AND SHIMURA’S CORRESPONDENCE13

are functions on X_{0}(27) satisfying Y^{2} = 4X^{3} − 27. One has

L(E/Q, s) = ∑_{n=1}^{∞} a_{n}n^{−s} = L(f, s),

where

f = ∑_{n=1}^{∞} a_{n}q^{n}

is the normalized (a_{1} = 1) generator of S_{2}(Γ_{0}(27), 1).

Let D be a square-free integer prime to 3 and Δ the conductor of the primitive Dirichlet character (\frac{D}{.} ). Then

L(E_{D}/Q, s) = ∑_{n=1}^{∞} a_{n} (\frac{D}{n} ) n^{−s} = L(f ⊗ (\frac{D}{·} ) , s),

where

f ⊗ (\frac{D}{·} ) = ∑_{n=1}^{∞} a_{n} (\frac{D}{n} ) q^{n} ∈ S_{2}(Γ_{0}(27Δ^{2}), 1).

Put Λ_{D}(s) = (27Δ^{2})^{s/2}(2π)^{−s}Γ(s)L(E_{D}/Q, s) . Then Λ_{D}(s) has a holomorphic continuation to C and satisfies the functional equation

Λ_{D}(s) = (\frac{−3}{|D|}) Λ_{D}(2 − s).

It follows that L(E_{D}/Q, 1) = 0 for |D| ≡ 2 (mod 3).

6. Shimura’s correspondence.

To compute critical values L_{D}(1) := L(E_{D}/Q, 1) for |D| ≡ 1 (mod 3), we shall use a theorem of Waldspurger [22],[23]. We must first find some cusp forms of weight 3/2 which are mapped to the form f under Shimura’s correspondence [19]. This is a correspondence between Hecke eigenforms F ∈ M_{k+1/2}(Γ_{0}(4N)) and f ∈ M_{2k}(Γ_{0}(N)) with the same Hecke algebra eigenvalues: T_{p^{2}}F = λ_{p}F, T_{p}f = λ_{p}f for all primes p prime to 4N.

We start with the cubic extension K′/K = Q(\sqrt{−3}, \sqrt[3]{2})/Q(\sqrt{−3}) and the cubic character χ : G(K′/K) ⟶ μ_{3}, which correspods via reciprocity law to the character a ⟼ (\frac{2}{a})_{3}, defined on ideals a in Z[ρ] prime to (3). According to [16, 7.2.1], ρ = Ind_{G(Q̄/K)}^{G(Q̄/Q)}χ is a dihedral representation of G(Q̄/Q) and the Artin L-function

L(ρ, s) = ∑_{n=1}^{∞} b_{n}n^{−s}

corresponds to a cusp form

g = ∑_{n=1}^{∞} b_{n}q^{n} ∈ S_{1}(Γ_{0}(108), (\frac{−3}{.})).

In fact,

g(z) = ∑_{α≡1 (3)} (\frac{2}{α})_{3} q^{Nα} = \frac{(θ_{1} − θ_{9})(3θ_{27} − θ_{3})}{4},

where θ_{a}(z) = ∑_{n∈Z} q^{an^{2}}.

Lemma 6.1. ([18, 2.16]) Let 0 ≠ F ∈ M_{k}(Γ_{0}(N), χ). Then

deg(div(F)) = \{(2g − 2) + ∑_{P}(1 − e_{P}^{−1}) + #(cusps)\}\frac{k}{2} ,

where g = genus of X_{0}(N) and P runs through all non-equivalent elliptic points of Γ_{0}(N) with orders e_{P}.

Corollary 6.2. For 0 ≠ F ∈ M_{k}(Γ_{0}(108), χ), deg(div(F)) = 18k.

Proof . Indeed, g = 10, #(cusps) = 18 and Γ_{0}(108) has no elliptic points. □

Proposition 6.3.

(1) The form g(z) = ∑_{n=1}^{∞} b_{n}q^{n} generates S_{1}(Γ_{0}(108), (\frac{−3}{.})). It has zeros of order one at all cusps and no other zeros.

(2) g(z) = η(6z)η(18z) = q ∏_{n=1}^{∞}(1 − q^{6n})(1 − q^{18n}).

Proof . (1) As (\frac{−3}{.}) is an odd character, any modular form belonging to M_{1}(Γ_{0}(108), (\frac{−3}{.})) has integral order at every irregular cusp. We conclude by Corollary 6.2.

(2) By (1), it is sufficient to prove that η_{6}η_{18} ∈ S_{1}(Γ_{0}(108), (\frac{−3}{.})). It is wellknown that \frac{η^{3}}{η_{3}} ∈ M_{1}(Γ_{0}(9), (\frac{−3}{.})) and it is easy to check that (\frac{η_{18}}{η_{6}} )^{2} is a function on X_{0}(108) (cf. [12]). It follows that η_{6}η_{18} = (\frac{η_{18}}{η_{6}} )^{2} \frac{η_{6}^{3}}{η_{18}} transforms like a form of weight one on Γ_{0}(108) with character (\frac{−3}{.}). This function is obviously holomorphic in the upper half plane and vanishes at all cusps, so it lies necessarily in S_{1}(Γ_{0}(108), (\frac{−3}{.})). □

CLASS NUMBERS OF QUADRATIC FIELDS AND SHIMURA’S CORRESPONDENCE15

Proposition 6.4. Let F_{−} = gθ_{3}, F_{+} = gθ_{9}, G_{−} = g \frac{3θ_{27}−θ_{3}}{2} , G_{+} = g \frac{θ_{1}−θ_{9}}{2} . Then (1) Both subspaces

V_{−} = CF_{−} ⊕ CG_{−} ⊆ S_{3/2}(Γ_{0}(108), 1)
V_{+} = CF_{+} ⊕ CG_{+} ⊆ S_{3/2}(Γ_{0}(108), (\frac{−3}{.}))

are invariant under the action of all Hecke operators T_{p^{2}} (p > 3).

(2) The forms F_{±}, G_{±} are eigenfunctions of all T_{p^{2}} (p > 3).

(3) Under Shimura’s correspondence F_{±} are mapped to f ∈ S_{2}(Γ_{0}(27), 1) G_{±} are mapped to f′ = q − q^{2} + q^{4} + 3q^{5} + ⋯ ∈ S_{2}(Γ_{0}(54), 1) (f′ corresponds to the curve 54E in the tables [24], p.117 ).

Proof . (1) The key point is the following

Lemma 6.5. Let 0 ≠ F ∈ M_{k/2}(Γ_{0}(108), χ) for some odd k. Then, for all primes p > 3, ord_{c}T_{p^{2}}(gF) ≥ 1 at every cusp c.

Proof of the Lemma. One has T_{p^{2}}(gF) = ∑ c_{i}(gF |α_{i}) for certain matrices α_{i} ∈ M_{2}(Z) with det(α_{i}) = p^{2}. Fix a cusp c of width h and α ∈ SL_{2}(Z) with α(c) = ∞. Let q = exp(2πiα(z)/h) be the local parameter at c. As g has integral order at c, the q-expansion of gF |α_{i}α^{−1} contains terms q^{e} with e = \frac{i}{p^{2}} + \frac{j}{4} for some i, j ∈ Z, i ≥ 1, j ≥ 0. The whole sum T_{p^{2}}(gF)|α^{−1} will contain only terms q^{e} with e = \frac{k}{4} , k ∈ Z. The lowest term must be q^{e} with e = \frac{i}{p^{2}} + \frac{j}{4} = \frac{k}{4} , which can happen only for p^{2}|i, hence i ≥ p^{2} and e ≥ 1. Lemma is proved.

We now return to the proof of Proposition 6.4. By Lemma 6.5, all T_{p^{2}} map V_{−} to g · M_{1/2}(Γ_{0}(108), (\frac{−3}{.})) and V_{+} to g · M_{1/2}(Γ_{0}(108), 1). According to [17],

M_{1/2}(Γ_{0}(108), (\frac{−3}{.})) = Cθ_{3} ⊕ Cθ_{27}
M_{1/2}(Γ_{0}(108), 1) = Cθ_{1} ⊕ Cθ_{9}.

The claim follows.

(2) According to Corollary 6.2, for any 0 ≠ F ∈ S_{3/2}(Γ_{0}(108), χ), ord_{∞}F ≤ 27 − 17/4 < 23. By checking all coefficients up to q^{22} we coclude that T_{5^{2}}F_{±} = 0, T_{5^{2}}G_{±} = 3G_{±}. As the algebra of Hecke operators is commutative, it follows

16 JAN NEKOVÁŘ

that Ker(T_{5^{2}}|_{V_{±}}) = CF_{±}, Ker((T_{5^{2}} − 3 · 1)|_{V_{±}}) = CG_{±} are invariant subspaces with respect to the action of all T_{p^{2}}.

(3) Shimura’s correspondence maps any of the functions F_{±}, G_{±} to some Hecke eigenform F ∈ M_{2}(Γ_{0}(54), 1). As all eigenvalues λ of T_{5^{2}} acting on Eisenstein series satisfy λ ≡ 1 (mod 5), F must be a cusp form. Inspection of the tables [24, p.117] shows that S_{2}(Γ_{0}(54), 1) is spanned by the following eigenforms:

f = q − 2q^{4} − q^{7} + 5q^{13} + 4q^{16} + …
f′ = q − q^{2} + q^{4} + 3q^{5} − q^{7} − q^{8} + …
f′′ = q + q^{2} + q^{4} − 3q^{5} − q^{7} + q^{8} + …

(they correspond to the curves 27B, 54E and 54A respectively, in the notation of [24, p.117]) with T_{5}f = 0, T_{5}f′ = 3f′, T_{5}f′′ = −3f′′, which concludes the proof. □

Put

e(n) = \{ \begin{matrix}1, & for n ≢ 5 (mod 8) \\ \frac{1}{3}, & for n ≡ 5 (mod 8) \end{matrix}

c′(n) = c(n) × \{ \begin{matrix}e(n), & for n < 0 \\ e(−3n), & for n > 0 \end{matrix} ,

where c(±n) are the coefficients of

F_{±} = ∑_{n=1}^{∞} c(±n)q^{n}.

If D is prime to 3, then c′(D) = c(D), unless D_{−} ≡ 5 (mod 8).

Theorem 6.6. Let D be a square-free integer prime to 3. Then

L_{D}(1) = L(E_{D}/Q, 1) = ΩΔ^{−1/2}c′(D)^{2} × \{ \begin{matrix}1, & for D < 0 \\ 3^{1/2}, & for D > 0 \end{matrix} ,

where Ω = \frac{1}{2π} Γ (\frac{1}{3})^{3} and Δ is the conductor of (\frac{D}{·} ).

Proof . According to [23, p.379], L_{D}(1) = A(D)|D|^{−1/2}c(D)^{2}, where A(D) depends only on D (mod 24) and sgn(D). It is sufficient, therefore, to find the values of A(D) e.g. for |D| = 7, 10, 13, 19, 22, 73. To do this, we make use of Stevens’ tables in [20, p.199-200]. Stevens tabulates the values of

Λ(D) = Δ^{1/2}Ω^{−1}L_{D}(1) × \{ \begin{matrix}1, & for D < 0 \\ 3^{1/2}, & for D > 0 \end{matrix} ,

CLASS NUMBERS OF QUADRATIC FIELDS AND SHIMURA’S CORRESPONDENCE17

as his curve Y^{2} = 4X^{3} − 27 has a real period Ω^{+} = Ω · 3^{−1/2} and an imaginary period Ω^{−} = Ω · i. Inspection of the tables shows that in almost all cases

Λ(D) = c(D)^{2} × \{ \begin{matrix}1, & for D < 0 \\ 3, & for D > 0 \end{matrix}

Exceptions occur for D ≡ \{ \begin{matrix}5 (mod 8), & for D < 0 \\ 1 (mod 8), & for D > 0 \end{matrix} , when c(D) is to be replaced by c(D)/3. □

7. Birch and Swinnerton-Dyer’s conjecture for E_{D}.

In this section we examine the rational factor of L_{D}(1). In the notation of Proposition 4.4, put

L_{D}^{∗}(1) = L_{D}(1) (∫_{E_{D}(R)} |ω_{min}|_{∞})^{−1} ∏_{p} c_{p}^{−1}.

Its value is conjecturaly given by \underline{Birch} \underline{and} \underline{Swinnerton}- \underline{Dyer’s} \underline{conjecture}:

L_{D}^{∗}(1) = \{ \begin{matrix}0, & for rkE_{D}(Q) > 0 \\ \frac{#\underline{III}_{D}}{(#E_{D}(Q)_{tors})^{2}} , & for rkE_{D}(Q) = 0 \end{matrix} ,

where \underline{III}_{D} = \underline{III}(E_{D}/Q) is the Tate-Šafarevič group of E_{D}.

Proposition 7.1. Let D be a square-free integer, |D| ≡ 1 (mod 3). Define α(D), β(D) by

2^{α(D)} = ∏_{p|D} c_{p} = ∏_{p|D,p≠2}(1 + #{x ∈ F_{p} | x^{3} = 2})
β(D) = \frac{α(D)}{2} + \{\frac{D − 1}{2}\}

Then β(D) = #\{p|D, p = x^{2} + 27y^{2}\} + \frac{1}{2}#\{p|D, p ≡ 2 (mod 3)\} is an integer and

L_{D}^{∗}(1) = (\frac{c′(D)}{2^{β(D)}})^{2}

is a rational square.

Proof . From the definitions of L_{D}^{∗}(1), α(D), β(D), Proposition 4.4 and Theorem 6.6 we get

ω_{min} = 2^{2β(D)−α(D)+1}(|D|/Δ)^{1/2}ω

18 JAN NEKOVÁŘ

and

2^{2β(D)}L_{D}(1) = 2^{2β(D)}ΩΔ^{−1/2}c′(D)^{2} × \{ \begin{matrix}1, & for D < 0 \\ 3^{1/2}, & for D > 0 \end{matrix}

= 2^{2β(D)+1}c_{3}(|D|/Δ)^{1/2}c′(D)^{2} ∫_{E_{D}(R)} |ω|_{∞}

= 2^{α(D)}c_{3}c′(D)^{2} ∫_{E_{D}(R)} |ω_{min}|_{∞}

= c′(D)^{2} ∏_{p} c_{p} ∫_{E_{D}(R)} |ω_{min}|_{∞} □

Proposition 7.2. Let D ≠ 1 be a square-free integer with |D| ≡ 1 (mod 3). Then

c(D) ≡ \{ \begin{matrix}−h(D_{−}) (mod 3), & for D_{−} ≢ 5 (mod 8) \\ 0 (mod 3), & for D_{−} ≡ 5 (mod 8) \end{matrix} ,

where h(n) denotes the class number of Q(\sqrt{n}) (recall that D_{−} = min(D, −3D) ).

Proof . As g ≡ −(θ_{1} − θ_{9})θ_{3} (mod 3),

c(±|D|) ≡ −N_{±}(|D|) (mod 3),

where

N_{+}(a) = #{(x, y, z) ∈ Z^{3} | x^{2} + 3y^{2} + 9z^{2} = a},
N_{−}(a) = #{(x, y, z) ∈ Z^{3} | x^{2} + 3y^{2} + 3z^{2} = a}.

The statement of the Proposition is a consequence of the following Lemma 7.3. If a > 1 is a square-free integer, a ≡ 1 (mod 3), then

N_{−}(a) = h(−a) × \{ \begin{matrix}4, & for a ≡ 1, 2 (mod 4) \\ 16, & for a ≡ 7 (mod 8) \\ 24, & for a ≡ 3 (mod 8) \end{matrix}

N_{+}(a) = h(−3a) × \{ \begin{matrix}1, & for a ≡ 2, 3 (mod 4) \\ 4, & for a ≡ 5 (mod 8) \\ 6, & for a ≡ 1 (mod 8) \end{matrix}

Sketch of the proof . We combine two facts:

(1) If Q = x^{2} + 3y^{2} + 3z^{2} or Q = x^{2} + 3y^{2} + 9z^{2}, then the genus of Q contains only one class of forms.

(2) Siegel’s formula for the average number of representstions of a number by classes in a given genus (see e.g. [11]).

CLASS NUMBERS OF QUADRATIC FIELDS AND SHIMURA’S CORRESPONDENCE19

Proof of (1) is a routine application of reduction theory (see e.g. [9]) and is omitted, as well as an explicit calculation of all terms in Siegel’s formula. □

Corollary 7.4. For any square-free integer D ≠ 1, c′(D) is an integer and L_{D}^{∗}(1) ∈ Z [\frac{1}{2}] .

Corollary 7.5. If D ≠ 1 is a square-free integer, |D| ≡ 1 (mod 3) and D_{−} ≢ 5 (mod 8), then

L_{D}^{∗}(1) ≢ 0 (mod 3) ⟺ the Selmer group S(3, E_{D}/Q) = 0.

Proof . This follows easily from Theorem 4.6, Proposition 7.1 and Proposition 7.2. □

Remark. A theorem of Rubin [14] implies that for a prime p > 3

L_{D}^{∗}(1) ≢ 0 (mod p) ⟹ the Selmer group S(p, E_{D}/Q) = 0.

Examination of tables of the coefficients c(D) suggests that a more general congruence

(7.1)  c′(D) ≡ −h(D_{−}) (mod 3)

holds for all squre-free D ≠ 1, |D| ≡ 1 (mod 3). Unfortunately, the above method fails for D_{−} ≡ 5 (mod 8), because in this case one must take into account also representations of |D| by forms x^{2} + 3y^{2} + 27z^{2}, x^{2} + 9y^{2} + 27z^{2}, whose genera contain several classes of forms. If (7.1) was proved, it would make the restriction D_{−} ≢ 5 (mod 8) in Corollary 7.5 unnecessary.

Proposition 7.6. If D is a square-free integer, |D| ≡ 1 (mod 3), then

c(D) ≡ c(−D) (mod 2) and
c(D) ≢ 0 (mod 2) ⟺ β(D) = 0.

Proof . Obviously F_{+} ≡ F_{−} ≡ g = ∑_{n=1}^{∞} b_{n}q^{n} (mod 2). If p is a prime, then

b_{p} = \{ \begin{matrix}0, & for p ≡ 2 (mod 3) \\ 2, & for p = x^{2} + 27y^{2} \\ −1, & for p ≠ x^{2} + 27y^{2}, p ≡ 1 (mod 3) \end{matrix} ,

20 JAN NEKOVÁŘ

or, equivalently, b_{p} = #{x ∈ F_{p} | x^{3} = 2}−1. Let |D| = p_{1} … p_{k}. As g is a Hecke eigenform, b_{n} is a multiplicative function, so

c(D) ≡ c(−D) ≡ b_{|D|} = b_{p_{1}} … b_{p_{k}} (mod 2)

and b_{|D|} ≢ 0 (mod 2) ⟺ no prime p|D is of the form p ≡ 2 (mod 3) or p = x^{2} + 27y^{2} ⟺ β(D) = 0. □ Corollary 7.7. If D is a squre-free integer, |D| ≡ 1 (mod 3) and c(D) ≢ 0 (mod 2), then L_{D}^{∗}(1) = c′(D)^{2} is an odd square and the Selmer group S(2, E_{D}/Q) = 0 vanishes.

Proof . In [2] Aigner proves that β(D) = 0 ⟹ S(2, E_{D}/Q) = 0. □

We conclude with a list of conjectural values of #\underline{III}_{D} for square-free |D| < 500 with L_{D}^{∗}(1) ≠ 0, 1 :

\begin{aligned}#\underline{III}_{D} = & \underline{4} : \\ & \underline{9} : \\ & \underline{16} : \\ & \underline{25} : \\ & \underline{36} : \\ & \underline{49} : \end{aligned} \begin{matrix}− 58, −82, −85, −142, −166, −235 \\ − 253, −301, −346, −391, −406, −445 \\ − 454, −457, −466 \\ 127, 166, 262, 298, 358, 403, 433 \\ 445, 466, 478 \\ − 61, −118, −139, −157, −199, −211 \\ − 214, −241, −247, −274, −277, −286 \\ − 331, −334, −367, −370, −379, −493 \\ 67, 79, 103, 139, 151, 181, 199, 238 \\ 247, 271, 322, 331, 337, 418, 427, 469 \\ − 478 \\ 451 \\ − 133, −313, −349, −469, −481 \\ 211, 259, 367 \\ − 373 \end{matrix}

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