Eisenstein ideal

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.