The Taylor-Wiles construction and multiplicity one

Abstract

Wiles’ proof [17] of the modularity of semistable elliptic curves over Q relies on a construction of Taylor and Wiles [16] showing that certain Hecke algebras are complete intersections. These Hecke algebras are defined by considering the action of Hecke operators on spaces of modular forms of “minimal level”, or equivalently, on homology groups or Jacobians of modular curves. Taylor and Wiles proceed, roughly speaking, by “patching” algebras arising from forms of different levels. One of the deep results used in their construction was the fact that the homology of the modular curve becomes a free module (of rank two) over the Hecke algebra upon localization at certain maximal ideals. This result, known as a “multiplicity one” result, is a generalization of a theorem of Mazur [11]. Its proof relied on the q-expansion principle of Deligne-Rapoport and Katz, and the comparison of mod ` Betti and de Rham cohomologies (see Sect. 2.1 of [17]). Multiplicity one was thought to be a crucial ingredient of the Taylor-Wiles construction as well as other parts of Wiles’ proof. The purpose of this paper is to explain how to alter the arguments of [16] and [17] so that multiplicity one results are a byproduct rather than an ingredient. The key conceptual change underlying this improvement is the following: Rather than prove that (after localization) the Hecke algebra can be identified with the universal deformation ring of a mod ` Galois representation, we prove that the homology of the modular curve is a free module over this deformation ring. To carry this out, we change the Taylor-Wiles construction1 by 1) “patching” the modules as well as the algebras, and 2) applying the Auslander-Buchsbaum