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

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
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