# Congruences for modular forms mod 2 and quaternionic $S$-ideal classes

## TL;DR

This paper establishes numerous congruences modulo 2 between elliptic and Hilbert modular forms with different Atkin--Lehner eigenvalues, using quaternionic $S$-ideal classes and sign distribution analysis.

## Contribution

It introduces new methods involving quaternionic $S$-ideal classes to prove simultaneous mod 2 congruences among modular forms with varying Atkin--Lehner eigenvalues.

## Key findings

- Many simultaneous mod 2 congruences among modular forms
- Distribution patterns of Atkin--Lehner signs among newforms
- Application of quaternionic $S$-ideal classes in modular form theory

## Abstract

We prove many simultaneous congruences mod 2 for elliptic and Hilbert modular forms among forms with different Atkin--Lehner eigenvalues. The proofs involve the notion of quaternionic $S$-ideal classes and the distribution of Atkin--Lehner signs among newforms.