The Taylor-Wiles construction and multiplicity one. (English) Zbl 0916.11037

Diamond, Fred

Invent. Math. 128, No. 2, 379-391 (1997).

The author revisits the Taylor construction, which is a key element in Wiles’ proof of the modularity of elliptic curves. He shows that one can dispense with the usual multiplicity one theorems derived from \(q\)-expansions, and even recover them with different proofs. This is derived from a refinement of the commutative algebra estimates of Wiles and Lenstra. As an application he treats Wiles-Taylor, and also an example of Shimura curves where no \(q\)-expansion is available.

Reviewer: G.Faltings (Bonn)

MSC:

11F33	Congruences for modular and \(p\)-adic modular forms
11G35	Varieties over global fields

Keywords:

multiplicity one; q-extension; Shimura curves; modularity of elliptic curves