ON RANKS OF QUADRATIC TWISTS OF A MORDELL CURVE

Published

May 13, 2022

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

Pub Type

Journals

Download Paper

https://doi.org/10.1007/s11139-022-00585-1

Local Download

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 September 17, 2024)