Central values of L -functions of cubic twists
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Mathematische Annalen
Central values of L-functions of cubic twists Eugenia Rosu1,2 Received: 23 April 2019 / Revised: 3 February 2020 © The Author(s) 2020
Abstract We are interested in finding for which positive integers D we have rational solutions for the equation x 3 + y 3 = D. The aim of this paper is to compute the value of the L-function L(E D , 1) for the elliptic curves E D : x 3 + y 3 = D. For the case of p prime p ≡ 1 mod 9, two formulas have been computed by Rodriguez-Villegas and Zagier. We have computed formulas that relate L(E D , 1) to the square of a trace of a modular function at a CM point. This offers a criterion for when the integer D is the sum of two rational cubes. Furthermore, when L(E D , 1) is nonzero we get a formula for the number of elements in the Tate–Shafarevich group and we show that this number is a √ square when D is a norm in Q[ −3]. Mathematics Subject Classification Primary 11G40 · 11F67; Secondary 14H52
1 Introduction In the current paper we are interested in finding which positive integers D can be written as the sum of two rational cubes: x 3 + y 3 = D, x, y ∈ Q.
(1)
Despite the simplicity of the problem, an elementary approach to solving the Diophantine equation fails. However, we can restate the problem in the language of elliptic curves. After making the equation homogeneous, we get the equation x 3 + y 3 = Dz 3 that has a rational point at ∞ = [1 : −1 : 0]. Moreover, after a change of coordinates
Communicated by Wei Zhang.
B
Eugenia Rosu [email protected]
1
Max Planck Institute for Mathematics, Bonn, Germany
2
Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA
123
E. Rosu z X = 12D x+y , Y = 36D x−y x+y the equation becomes:
E D : Y 2 = X 3 − 432D 2 , which defines an elliptic curve over Q written in its Weierstrass affine form. Thus the problem reduces to finding if the group of rational points E D (Q) of the elliptic curve E D is non-trivial. We assume D cube free and D = 1, 2 throughout the paper. In this case E D (Q) has trivial torsion (see [27]), thus (1) has a solution iff E D (Q) has positive rank. From the Birch and Swinnerton-Dyer (BSD) conjecture, this is conjecturally equivalent to the vanishing of L(E D , 1). Without assuming BSD, from the work of Coates–Wiles [3] (or more generally Gross–Zagier [9] and Kolyvagin [17]), when L(E D , 1) = 0 the rank of E D (Q) is 0, thus we have no rational solutions in (1). In the case of prime numbers, Sylvester conjectured that we have solutions in (1) in the case of D ≡ 4, 7, 8 mod 9. In the cases of D prime with D ≡ 2, 5 mod 9, D is not the sum of two cubes. This follows from a 3-descent argument (given in the 19th century by Sylvester, Lucas and Pepin). We define the invariant SD =
L(E D , 1) , c3D Ω D
√ 3 3 Γ 13 is the real period and c3D = p|3D c p is the product of where Ω D = √ 3 6π D the Tamagawa numbers c p corresponding to the elliptic curve E D at the unramified places p|3D. The definition is made such that in the case of L(E D , 1) = 0 we expect to get from th
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