Root numbers of elliptic curves

Master thesis: Nikolaos Tsopanidis

Abstract:
In the present paper we examine which numbers can be written as a sum of 2 rational cubes, meaning searching for solutions for the equation x^3 + y^3 = D in Q. At first we work for D a prime number and then for D an integer. In both cases we arrive to a formula for a certain number called the root number which can only take the values 1. We do so by proving the sextic reciprocity law in the Eisenstein integers and utilizing some results from the theory of elliptic curves and Tate's thesis. Finally we use a famous conjecture called the Birch and Swinnerton-Dyer conjecture which has not been proven yet, but there are overwhelming indications for its truth. The conjecture predicts that a certain number, prime or integer for our 2 separate cases, can be written as a sum of 2 cubes as long as we know that the formula mentioned above gives a root number equal to -1

Supervisor: Ian Kiming
Censor: Tom Høholdt