UC Berkeley

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 thesis is to compute the value of the L-function L(E_D, 1), for E_D: x^3+y^3=D. For the case of D prime D=1 mod 9, two formulas have been computed by Rodriguez-Villegas and Zagier. We have computed several formulas that relate $L(E_D, 1)$ to the 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 of E_D.