Higher congruences between newforms and Eisenstein series of squarefree level

Abstract

Let p ≥ 5 be prime. For squarefree level N > 6, 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 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.