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ON RANKS OF QUADRATIC TWISTS OF A MORDELL CURVE

Published

Author(s)

Abhishek Juyal, Dustin Moody, Bidisha Roy

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 J. Silverman that there are infinitely many primes $p$ for which $E_p(\mathbbQ})$ has a positive rank as well as infinitely many primes $q$ for which $E_q(\mathbbQ})$ has rank $0$. We show, assuming the Parity Conjecture, that Silverman's conjecture is true for this family of quadratic twists.
Citation
The Ramanujan Journal
Volume
59

Keywords

Twists of curves, Mordell curves, Selmer groups, rank of elliptic curves, root numbers

Citation

Juyal, A. , Moody, D. and Roy, B. (2022), ON RANKS OF QUADRATIC TWISTS OF A MORDELL CURVE, The Ramanujan Journal, [online], https://doi.org/10.1007/s11139-022-00585-1, https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=934636 (Accessed May 19, 2024)

Issues

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Created May 13, 2022, Updated May 10, 2024