Higher Eisenstein elements, higher Eichler formulas and rank of Hecke algebras

TL;DR

This paper investigates the rank of Hecke algebras at Eisenstein ideals for modular forms, introduces higher Eisenstein elements, and provides criteria for rank bounds, extending classical formulas in characteristic p.

Contribution

It develops the theory of higher Eisenstein elements and applies it to determine the rank of Hecke algebras, partially answering Mazur's question.

Findings

Explicit criterion for when the rank g_p is at least 3.

Construction of higher Eisenstein elements in four Hecke modules.

Generalizations of Eichler mass formula in characteristic p.

Abstract

Let $N$ and $p$ be primes such that $p$ divides the numerator of $\frac{N-1}{12}$. In this paper, we study the rank $g_p$ of the completion of the Hecke algebra acting on cuspidal modular forms of weight $2$ and level $\Gamma_0(N)$ at the $p$-maximal Eisenstein ideal. We give in particular an explicit criterion to know if $g_p \geq 3$, thus answering partially a question of Mazur. In order to study $g_p$, we develop the theory of \textit{higher Eisenstein elements}, and compute the first few such elements in four different Hecke modules. This has applications such as generalizations of the Eichler mass formula in characteristic $p$.