Central values of L-functions of cubic twists

Abstract

AbstractWe are interested in finding for which positive integers D we have rational solutions for the equation $$x^3+y^3=D.$$ x 3 + y 3 = D . The aim of this paper is to compute the value of the L-function $$L(E_D, 1)$$ L ( E D , 1 ) for the elliptic curves $$E_D: x^3+y^3=D$$ E D : x 3 + y 3 = D . For the case of p prime $$p\equiv 1\mod 9$$ p ≡ 1 mod 9 , two formulas have been computed by Rodriguez-Villegas and Zagier. We have computed formulas that relate $$L(E_D, 1)$$ 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 $$L(E_D, 1)$$ 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 $${\mathbb {Q}}[\sqrt{-3}]$$ Q [ - 3 ] .