On the rank of cubic and quartic twists of elliptic curves by primes

Abstract

We show that there are infinitely many primes p for which the cubic twist y2=x3+16p2 (respectively the quartic twist y2=x3−px) has positive Mordell-Weil rank by playing off the structure of the Q-rational torsion points again the projective geometry of these twists. A key ingredient to both proofs are sieve results of Heath-Brown and Moroz, and Friedlander and Iwaniec, on primes represented by cubic and quartic polynomials. As a consequence we show that there are infinitely many primes congruent to 1(mod9) and respectively −1(mod9) that can be written as a sum of two rational cubes. We show that there are infinitely many primes p for which the cubic twist y2=x3+16p2 (respectively the quartic twist y2=x3−px) has positive Mordell-Weil rank by playing off the structure of the Q-rational torsion points again the projective geometry of these twists. A key ingredient to both proofs are sieve results of Heath-Brown and Moroz, and Friedlander and Iwaniec, on primes represented by cubic and quartic polynomials. As a consequence we show that there are infinitely many primes congruent to 1(mod9) and respectively −1(mod9) that can be written as a sum of two rational cubes.