Modular Hecke algebra ($\mathbb{T}$)

A modular Hecke algebra is the algebra generated by Hecke operators acting on a specified space of modular forms, with level, weight and coefficient ring fixed. Its Eisenstein ideal encodes congruences with Eisenstein eigenvalues. This arithmetic algebra should not be confused with the generic Hecke algebra of a Coxeter group.

The Eisenstein ideal in a modular Hecke algebra imposes the eigenvalue relations of an Eisenstein series, such as $T_{\ell }-(1+\ell )$ in an appropriate weight-two trivial-character setting. Congruences modulo this ideal connect modular forms and class-field constructions in the Mazur-Wiles theorem. Other weights and characters require adjusted eigenvalues.