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Modular Forms, Elliptic Curves, and Modular Curves.- Modular Curves as Riemann Surfaces.- Dimension Formulas.- Eisenstein Series.- Hecke Operators.- Jacobians and Abelian Varieties.- Modular Curves… Expand
In this paper, building on work of Wiles [Wi] and of Wiles and one of us (R.T.) [TW], we will prove the following two theorems (see §2.2). Theorem A. If E/Q is an elliptic curve, then E is modular.… Expand
The authors would like to give special thanks to N. Boston, K. Buzzard, and B. Conrad for providing so much valuable feedback on earlier versions of this paper. They are also grateful to A. Agboola,… Expand
In 1987 Serre conjectured that any mod ' two-dimensional irre- ducible odd representation of the absolute Galois group of the rationals came from a modular form in a precise way. We present a… Expand
Wiles’ proof  of the modularity of semistable elliptic curves over Q relies on a construction of Taylor and Wiles  showing that certain Hecke algebras are complete intersections. These Hecke… Expand
where is the reduction of p. These results were subject to hypotheses on the local behavior of p at X, i.e., the restriction of p to a decomposition group at X, and to irreducibility hypotheses on… Expand