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

TL;DR

This paper proves Watkins' conjecture for quadratic twists of elliptic curves with rational 2-torsion and prime power conductor, and provides a lower bound for the congruence number of certain elliptic curves.

Contribution

It establishes Watkins' conjecture in a new class of elliptic curves and derives a lower bound for the congruence number for specific elliptic curves.

Findings

Watkins' conjecture holds for quadratic twists with rational 2-torsion and prime power conductor.

A lower bound for the congruence number is provided for curves of the form y^2=x^3-dx.

The results extend the understanding of the relationship between rank and modular degree in elliptic curves.

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.