Hecke algebras associated to $\Lambda$-adic modular forms

TL;DR

This paper investigates the structure of $p$-adic Hecke algebras linked to modular forms, establishing conditions under which these algebras are Gorenstein based on ideal class groups and conjectures.

Contribution

It connects the Gorenstein property of Eisenstein components of $p$-adic Hecke algebras to ideal class groups and Sharifi's conjecture, providing new criteria for algebraic structure.

Findings

Gorenstein property implies trivial plus-part of a certain ideal class group.

The triviality condition is also sufficient if Sharifi's conjecture holds.

Establishes a link between algebraic properties and class group structure in modular forms.

Abstract

We show that if an Eisenstein component of the $p$-adic Hecke algebra associated to modular forms is Gorenstein, then it is necessary that the plus-part of a certain ideal class group is trivial. We also show that this condition is sufficient whenever a conjecture of Sharifi holds.