Twists of elliptic curves of rank at least four

Summary

Abstract

We give infinite families of elliptic curves over ℚ such that each curve has infinitely many non-isomorphic quadratic twists of rank at least 4. Assuming the Parity Conjecture, we also give elliptic curves over ℚ with infinitely many non-isomorphic quadratic twists of odd rank at least 5.

Introduction

Mestre showed that every elliptic curve over ℚ has infinitely many (non-isomorphic) quadratic twists of rank at least 2 over ℚ, and he gave several infinite families of elliptic curves over ℚ with infinitely many (non-isomorphic) quadratic twists of rank at least 3. Further, he stated that if E is an elliptic curve over ℚ with torsion subgroup isomorphic to ℤ/8ℤ × ℤ/2ℤ, then there are infinitely many (non-isomorphic) quadratic twists of E with rank at least 4 over ℚ.

In this paper (Theorems 3.2 and 3.6) we give additional infinite families of elliptic curves over ℚ with infinitely many (non-isomorphic) quadratic twists of rank at least 4. The family of elliptic curves in Theorem 3.2 is parametrized by the projective line. The family of elliptic curves in Theorem 3.6 is parametrized by an elliptic curve of rank one. In both cases, the twists are parametrized by an elliptic curve of rank at least one.

In addition, we find elliptic curves over ℚ that, assuming the Parity Conjecture, have infinitely many (non-isomorphic) quadratic twists of odd rank at least 5 (see Theorem 5.1 and Corollary 5.2). The proof relies on work of Rohrlich.