Level raising mod 2 and arbitrary 2-Selmer ranks

TL;DR

This paper proves a level raising mod 2 theorem for elliptic curves over Q, generalizing previous results and analyzing 2-Selmer groups, showing their ranks can vary arbitrarily within level raising families.

Contribution

It introduces a new level raising mod 2 theorem for elliptic curves, extending prior work and providing insights into 2-Selmer group behavior.

Findings

The 2-Selmer rank can be arbitrarily large in level raising families.

The theorem explains sign phenomena differences for  compared to odd primes.

The results generalize Ribet and Diamond-Taylor theorems.

Abstract

We prove a level raising mod $\ell=2$ theorem for elliptic curves over $\mathbb{Q}$. It generalizes theorems of Ribet and Diamond-Taylor and also explains different sign phenomena compared to odd $\ell$. We use it to study the 2-Selmer groups of modular abelian varieties with common mod 2 Galois representation. As an application, we show that the 2-Selmer rank can be arbitrary in level raising families.