Fred Diamond

On the Modularity of Elliptic Curves Over Q: Wild 3-Adic Exercises

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. Theorem B. If ρ : Gal(Q/Q) → GL2(F5) is an irreducible continuous representation with cyclotomic determinant, then ρ is modular. We will first remind the reader of the content of these results and then briefly outline the method of proof. If N is a positive integer then we let Γ1(N) denote the subgroup of SL2(Z) consisting of matrices that modulo N are of the form ( 1 ∗ 0 1 ) .

An Extension of Wiles’ Results

Suppose that E is an elliptic curve defined over Q. We wish to prove that E is modular, or equivalently, that the associated l-adic representation

Elliptic curves, Hilbert modular forms and Galois deformations

\rho E,\ell :G_Q \to Aut(T_\ell (E)) \cong GL_2 (Z_\ell ).

Minimal weights of Hilbert modular forms in characteristic p

is modular for some primel.

Irreducible modular representations of the Borel subgroup of GL2(Qp)

Let K be a finite unramified extension of Q_p. We parametrize the (phi, Gamma)-modules corresponding to reducible two-dimensional mod p representations of G_K and characterize those which have reducible crystalline lifts with certain Hodge-Tate weights.

Effective local Langlands correspondence

Part I: Galois Deformations.- On p-adic Galois Representations.- Deformations of Galois Representations.- Part II: Hilbert Modular Forms.- Arithmetic Aspects of Hilbert Modular Forms and Varieties.- Explicit Methods for Hilbert Modular Forms.- Part III: Elliptic Curves.- Notes on the Parity Conjecture.

Conditional results on the birational section conjecture over small number fields

A generalization of Serres Conjecture asserts that if

Geometry of the fundamental lemma

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