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Displaying similar documents to “Coupling of universal monodromy representations of Galois-Teichmüller modular groups.”

On the equation a³ + b³ⁿ = c²

Michael A. Bennett, Imin Chen, Sander R. Dahmen, Soroosh Yazdani (2014)

Acta Arithmetica

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We study coprime integer solutions to the equation a³ + b³ⁿ = c² using Galois representations and modular forms. This case represents perhaps the last natural family of generalized Fermat equations descended from spherical cases which is amenable to resolution using the so-called modular method. Our techniques involve an elaborate combination of ingredients, ranging from ℚ-curves and a delicate multi-Frey approach, to appeal to intricate image of inertia arguments.

Computations of Galois representations associated to modular forms of level one

Peng Tian (2014)

Acta Arithmetica

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We propose an improved algorithm for computing mod ℓ Galois representations associated to a cusp form f of level one. The proposed method allows us to explicitly compute the case with ℓ = 29 and f of weight k = 16, and the cases with ℓ = 31 and f of weight k = 12,20,22. All the results are rigorously proved to be correct. As an example, we will compute the values modulo 31 of Ramanujan's tau function at some huge primes up to a sign. Also we will give an improved...

On realizability of p-groups as Galois groups

Michailov, Ivo M., Ziapkov, Nikola P. (2011)

Serdica Mathematical Journal

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2000 Mathematics Subject Classification: 12F12, 15A66. In this article we survey and examine the realizability of p-groups as Galois groups over arbitrary fields. In particular we consider various cohomological criteria that lead to necessary and sufficient conditions for the realizability of such a group as a Galois group, the embedding problem (i.e., realizability over a given subextension), descriptions of such extensions, automatic realizations among p-groups, and related...

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

B. Mazur (1997)

Collectanea Mathematica

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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...

Modularity of Galois representations

Chris Skinner (2003)

Journal de théorie des nombres de Bordeaux

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This paper is essentially the text of the author’s lecture at the 2001 Journées Arithmétiques. It addresses the problem of identifying in Galois-theoretic terms those two-dimensional, p -adic Galois representations associated to holomorphic Hilbert modular newforms.