Abstract
In this article, we consider the quadratic twists of the Mordell curve \(E:y^2=x^3-1\). For a square-free integer k, the quadratic twist of E is given by \(E_k:y^2=x^3-k^3.\) We prove that there exist infinitely many k for which the rank of \(E_k\) is 0, by modifying existing techniques. Moreover, using simple tools, we produce precise values of k for which the rank of \(E_k\) is 0. We also construct an infinite family of curves \(\{ E_k \}\) such that the rank of each \(E_k\) is positive. It was conjectured by Silverman that there are infinitely many primes p for which \(E_p({\mathbb {Q}})\) has a positive rank as well as infinitely many primes q for which \(E_q({\mathbb {Q}})\) 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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Abhishek Juyal research is supported by NBHM post-doctoral fellowship (54603/2021/NBHM)
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Juyal, A., Moody, D. & Roy, B. On ranks of quadratic twists of a Mordell curve. Ramanujan J 59, 31–50 (2022). https://doi.org/10.1007/s11139-022-00585-1
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DOI: https://doi.org/10.1007/s11139-022-00585-1