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Displaying similar documents to “Computing the number of certain Galois representations mod p ”

Congruences between modular forms and lowering the level mod ℓ n

Luis Dieulefait, Xavier Taixés i Ventosa (2009)

Journal de Théorie des Nombres de Bordeaux

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In this article we study the behavior of inertia groups for modular Galois mod ℓ n representations and in some cases we give a generalization of Ribet’s lowering the level result (cf. []).

On a conjecture of Watkins

Neil Dummigan (2006)

Journal de Théorie des Nombres de Bordeaux

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Watkins has conjectured that if R is the rank of the group of rational points of an elliptic curve E over the rationals, then 2 R divides the modular parametrisation degree. We show, for a certain class of E , chosen to make things as easy as possible, that this divisibility would follow from the statement that a certain 2 -adic deformation ring is isomorphic to a certain Hecke ring, and is a complete intersection. However, we show also that the method developed by Taylor, Wiles and others,...

Hopf-Galois module structure of tame biquadratic extensions

Paul J. Truman (2012)

Journal de Théorie des Nombres de Bordeaux

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In [] we studied the nonclassical Hopf-Galois module structure of rings of algebraic integers in some tamely ramified extensions of local and global fields, and proved a partial generalisation of Noether’s theorem to this setting. In this paper we consider tame Galois extensions of number fields L / K with group G ≅ C 2 × C 2 and study in detail the local and global structure of the ring of integers 𝔒 L as a module over its associated order 𝔄 H in each of the Hopf algebras H giving a nonclassical Hopf-Galois...

The equation x 2 n + y 2 n = z 5

Michael A. Bennett (2006)

Journal de Théorie des Nombres de Bordeaux

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We show that the Diophantine equation of the title has, for n > 1 , no solution in coprime nonzero integers x , y and z . Our proof relies upon Frey curves and related results on the modularity of Galois representations.