Eisenstein congruences at prime-square level and an application to class numbers

Abstract: In Mazur's seminal work on the Eisenstein ideal, he showed that when $N$ and $p > 3$ are primes, there is a weight $2$ cusp form of level $N$ congruent to the unique weight $2$ Eisenstein series of level $N$ if and only if $N = 1$ mod $p$. Calegari--Emerton, Lecouturier, and Wake--Wang-Erickson have work that relates these cuspidal-Eisenstein congruences to the $p$-part of the class group of $\mathbb{Q}(N^{1/p})$. Calegari observed that when $N = -1$ mod $p$, one can use Galois cohomology and some ideas of Wake--Wang-Erickson to show that $p$ divides the class group of $\mathbb{Q}(N^{1/p})$. He asked whether there is a way to directly construct the relevant degree $p$ everywhere unramified extension of $\mathbb{Q}(N^{1/p})$ in this case. After discussing some of this background, I will report of work in progress with Preston Wake in which we give a positive answer to this question using cuspidal-Eisenstein congruences at prime-square level.

algebraic geometrynumber theory

Audience: researchers in the topic

UCSB Seminar on Geometry and Arithmetic