Watkins's conjecture for quadratic twists of Elliptic Curves with Prime Power Conductor

Jerson Caro

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arXiv · 2206.10008

Watkins's conjecture for quadratic twists of Elliptic Curves with Prime Power Conductor

Jerson Caro

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Abstract

Watkins' conjecture asserts that the rank of an elliptic curve is upper bounded by the $2$-adic valuation of its modular degree. We show that this conjecture is satisfied when $E$ is any quadratic twist of an elliptic curve with rational $2$-torsion and prime power conductor. Furthermore, we give a lower bound of the congruence number for elliptic curves of the form $y^2=x^3-dx$, with $d$ a biquadratefree integer.

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Jerson Caro. 2022-06-20. Watkins's conjecture for quadratic twists of Elliptic Curves with Prime Power Conductor. https://arxiv.org/abs/2206.10008

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