Central values of L-functions of cubic twists

Eugenia Rosu
2020 Mathematische Annalen  
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
more » ... 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 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
doi:10.1007/s00208-020-02018-0 fatcat:3q4yjshyunbgbgjqljirvcbkmi