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A combinatorial interpretation of Serre's conjecture on modular Galois representations

Adriaan Herremans (2003)

Annales de l’institut Fourier

We state a conjecture concerning modular absolutely irreducible odd 2-dimensional representations of the absolute Galois group over finite fields which is purely combinatorial (without using modular forms) and proof that it is equivalent to Serre’s strong conjecture. The main idea is to replace modular forms with coefficients in a finite field of characteristic p , by their counterparts in the theory of modular symbols.

An explicit formula for the Hilbert symbol of a formal group

Floric Tavares Ribeiro (2011)

Annales de l’institut Fourier

A Brückner-Vostokov formula for the Hilbert symbol of a formal group was established by Abrashkin under the assumption that roots of unity belong to the base field. The main motivation of this work is to remove this hypothesis. It is obtained by combining methods of ( ϕ , Γ )-modules and a cohomological interpretation of Abrashkin’s technique. To do this, we build ( ϕ , Γ )-modules adapted to the false Tate curve extension and generalize some related tools like the Herr complex with explicit formulas for the...

An infinite ferm in the universal deformation space of Galois representations.

B. Mazur (1997)

Collectanea Mathematica

I hope this article will be helpful to people who might want a quick overview of how modular representations fit into the theory of deformations of Galois representations. There is also a more specific aim: to sketch a construction of a point-set topological'' configuration (the image of an infinite fern'') which emerges from consideration of modular representations in the universal deformation space of all Galois representations. This is a configuration hinted previously, but now, thanks to some...

Automorphic realization of residual Galois representations

Robert Guralnick, Michael Harris, Nicholas M. Katz (2010)

Journal of the European Mathematical Society

We show that it is possible in rather general situations to obtain a finite-dimensional modular representation ρ of the Galois group of a number field F as a constituent of one of the modular Galois representations attached to automorphic representations of a general linear group over F , provided one works “potentially.” The proof is based on a close study of the monodromy of the Dwork family of Calabi–Yau hypersurfaces; this in turn makes use of properties of rigid local systems and the classification...

Automorphy for some l-adic lifts of automorphic mod l Galois representations. II

Richard Taylor (2008)

Publications Mathématiques de l'IHÉS

We extend the results of [CHT] by removing the ‘minimal ramification’ condition on the lifts. That is we establish the automorphy of suitable conjugate self-dual, regular (de Rham with distinct Hodge–Tate numbers), l-adic lifts of certain automorphic mod l Galois representations of any dimension. The main innovation is a new approach to the automorphy of non-minimal lifts which is closer in spirit to the methods of [TW] than to those of [W], which relied on Ihara’s lemma.

Automorphy for some l-adic lifts of automorphic mod l Galois representations

Laurent Clozel, Michael Harris, Richard Taylor (2008)

Publications Mathématiques de l'IHÉS

We extend the methods of Wiles and of Taylor and Wiles from GL2 to higher rank unitary groups and establish the automorphy of suitable conjugate self-dual, regular (de Rham with distinct Hodge–Tate numbers), minimally ramified, l-adic lifts of certain automorphic mod l Galois representations of any dimension. We also make a conjecture about the structure of mod l automorphic forms on definite unitary groups, which would generalise a lemma of Ihara for GL2. Following Wiles’ method we show that this...

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