On ranks of quadratic twists of a Mordell curve

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Research Article

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Ramanujan Journal


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.

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