Companion forms and the structure of p-adic Hecke algebras

TL;DR

This paper investigates the structure of p-adic Hecke algebras related to modular forms, establishing conditions for their Gorenstein property and exploring implications for Iwasawa theory.

Contribution

It provides a new criterion for the Gorenstein property of Eisenstein components of p-adic Hecke algebras using companion forms and offers a different approach from existing work.

Findings

Gorenstein property linked to one-dimensionality of modular forms space

Numerical criterion for one-dimensionality of modular forms space

Connections established between Hecke algebra structure and Iwasawa theory

Abstract

We study the structure of the Eisenstein component of Hida's ordinary p-adic Hecke algebra attached to modular forms, in connection with the companion forms in the space of modular forms (mod p). We show that such an algebra is a Gorenstein ring if certain space of modular forms (mod p) having companions is one-dimensional; and also give a numerical criterion for this one-dimensionality. This in part overlaps with a work of Skinner and Wiles; but our method, based on a work of Ulmer, is totally different. We then consider consequences of the above mentioned Gorenstein property. We especially discuss the connection with the Iwasawa theory.