title: Congruences between modular forms: raising the level and dropping Euler factors.
abstract: We discuss the relationship among certain generalizations of results of Hida, Ribet, and Wiles on congruences between modular forms. Hida's result accounts for congruences in terms of the value of an L-function, and Ribet's result is related to the behavior of the period that appears there. Wiles' theory leads to a class number formula relating the value of the L-function to the size of a Galois cohomology group. The behavior of the period is used to deduce that a formula at "nonminimal level" is obtained from one at "minimal level" by dropping Euler factors from the L-function.
authors.affiliation: Department of Pure Mathematics and Mathematical Statistics, 16 Mill Lane, University of Cambridge, Cambridge CB2 1SB, United Kingdom.
authors.forename: F
authors.initials: F
authors.lastname: Diamond