ON TWISTS OF THE FERMAT CUBIC x³ + y³ = 2

Abstract

We consider the Fermat elliptic curve E 2 : x 3 + y 3 = 2 and prove (using descent methods) that its quadratic twists have rank zero for a positive proportion of squarefree integers with fixed number of prime divisors. We also prove similar result for rank zero cubic twists of this curve. Then we present detailed description of rank zero quadratic and cubic twists of E 2 by primes and by products of two primes. We also consider twists of Jacobians of Fermat curves x 5 + y 5 = m and distribution of their root numbers.