Rank parity for congruent supersingular elliptic curves

TL;DR

This paper extends the understanding of rank parity in elliptic curves by proving an analogous result for supersingular primes, complementing prior work on ordinary primes.

Contribution

It establishes a new rank parity result for elliptic curves with supersingular reduction, filling a gap in the existing theory.

Findings

Proves rank parity for supersingular elliptic curves with isomorphic p-torsion modules.

Complements existing results for ordinary primes.

Enhances understanding of elliptic curve ranks in different reduction types.

Abstract

A recent paper of Shekhar compares the ranks of elliptic curves $E_1$ and $E_2$ for which there is an isomorphism $E_1[p] \simeq E_2[p]$ as $\mathrm{Gal}(\bar{\mathbf{Q}}/\mathbf{Q})$-modules, where $p$ is a prime of good ordinary reduction for both curves. In this paper we prove an analogous result in the case where $p$ is a prime of good supersingular reduction.