The level 1 weight 2 case of Serre’s conjecture. (English) Zbl 1171.11032

Dieulefait, Luis

Rev. Mat. Iberoam. 23, No. 3, 1115-1124 (2007).

In this work, the author proves Serre’s conjecture for the case of Galois representations of Serre’s weight 2 and level 1; equivalently he proves the non existence of odd, 2-dimensional, irreducible representations of the absolute Galois group of \(Q\) with values in a finite field of odd characteristic \(p>3\), in the case of Serre’s weight 2 and level 1. He uses the potential modularity results of Taylor and he applies a generalization of Ribet’s level-lowering result to the case of Hilbert modular forms. Then he uses a Galois descent argument and properties of the universal deformation rings, and to conclude the proof he applies his result, predicted by the Fontaine-Mazur conjecture, about the non-existence of \(p\)-adic Barsotti-Tate conductor 1 Galois representation.
The main new idea introduced in this article is the use of potential modularity to obtain from lowering the level of Hilbert modular forms the existence of minimal \(p\)-adic deformation of certain residual Galois representations.

Reviewer: Miriam Ciavarella (Torino)

MSC:

11F80	Galois representations
11F11	Holomorphic modular forms of integral weight
11F41	Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces

Keywords:

Galois representations; modular forms

References:

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