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Central values of L-functions of cubic twists

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Roşu,  Eugenia
Max Planck Institute for Mathematics, Max Planck Society;

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Roşu, E. (2020). Central values of L-functions of cubic twists. Mathematische Annalen, 378(3-4), 1327-1370. doi:10.1007/s00208-020-02018-0.


Cite as: http://hdl.handle.net/21.11116/0000-0007-0352-9
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 $p\equiv 1\mod 9$, 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 $\mathbb{Q}[\sqrt{-3}]$.