On ranks of quadratic twists of a Mordell curve

Article Type

Research Article

Publication Title

Ramanujan Journal

Abstract

In this article, we consider the quadratic twists of the Mordell curve E: y2= x3- 1. For a square-free integer k, the quadratic twist of E is given by Ek: y2= x3- k3. We prove that there exist infinitely many k for which the rank of Ek is 0, by modifying existing techniques. Moreover, using simple tools, we produce precise values of k for which the rank of Ek is 0. We also construct an infinite family of curves { Ek} such that the rank of each Ek is positive. It was conjectured by Silverman that there are infinitely many primes p for which Ep(Q) has a positive rank as well as infinitely many primes q for which Eq(Q) has rank 0. We show, assuming the Parity Conjecture that Silverman’s conjecture is true for this family of quadratic twists.

First Page

31

Last Page

50

DOI

10.1007/s11139-022-00585-1

Publication Date

9-1-2022

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