# Higher congruences between newforms and Eisenstein series of squarefree level

## TL;DR

This paper investigates the depth of Eisenstein congruences for modular forms of squarefree level, linking it to Bernoulli numbers and the Eisenstein ideal's properties, with applications to algebraic structures.

## Contribution

It establishes bounds on Eisenstein congruence depth using Bernoulli numbers and explores the non-principality of Eisenstein ideals through explicit examples and algebraic applications.

## Key findings

- Bound the depth of Eisenstein congruences using Bernoulli numbers.
- Identify cases where the Eisenstein ideal is not locally principal.
- Provide explicit computations and algebraic applications related to multiplicities.

## Abstract

Let $p\geq 5$ be prime. For elliptic modular forms of weight 2 and level $\Gamma_0(N)$ where $N>6$ is squarefree, we bound the depth of Eisenstein congruences modulo $p$ (from below) by a generalized Bernoulli number with correction factors and show how this depth detects the local non-principality of the Eisenstein ideal. We then use admissibility results of Ribet and Yoo to give an infinite class of examples where the Eisenstein ideal is not locally principal. Lastly, we illustrate these results with explicit computations and give an interesting commutative algebra application related to Hilbert--Samuel multiplicities.