Central values of $L$-functions of cubic twists

TL;DR

This paper computes special values of L-functions for elliptic curves related to sums of two rational cubes, providing criteria for rational solutions and insights into the Tate-Shafarevich group structure.

Contribution

It relates L-values of elliptic curves to modular functions at CM points, extending previous formulas and offering new criteria for rational cube sums.

Findings

Formulas relating L(E_D, 1) to traces of modular functions at CM points

Criteria for D being a sum of two rational cubes based on L-values

Demonstration that the Tate-Shafarevich group's size is a perfect square when D is a norm

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 $\mathbb{Q}[\sqrt{-3}]$.