Ring-Theoretic Properties of Certain Hecke Algebras

The purpose of this article is to provide a key ingredient of W2 by establishing that certain minimal Hecke algebras considered there are complete intersections. As is recorded in W2, a method going back to Mazur M allows one to show that these algebras are Gorenstein, but for the complete intersection property a new approach is required. The methods of this paper are related to those of Chapter 3 of W2. The methods of Section 3 of this paper are based on a previous approach of one of us (A.W.). We would like to thank Henri Darmon, Fred Diamond and Gerd Faltings for carefully reading the first version of this article. Gerd Faltings has also suggested a simplification of our argument as well as of the argument of Chapter 3 of W2 and we would like to thank him for allowing us to reproduce these in the appendix to this paper. R. T. would like to thank A. W. for his invitation to collaborate and for sharing his many insights into the questions considered. R. T. would also like to thank Princeton University, Universite de Paris 7 and Harvard University for their hospitality during this collaboration. A. W. was supported by an NSF grant.