Class 2 Galois representations of Kummer type.
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 uper bound on...
Using the link between Galois representations and modular forms established by Serre’s Conjecture, we compute, for every prime , a lower bound for the number of isomorphism classes of Galois representation of on a two–dimensional vector space over which are irreducible, odd, and unramified outside .
Let be a positive integer divisible by 4, a prime, an elliptic cuspidal eigenform (ordinary at ) of weight , level 4 and non-trivial character. In this paper we provide evidence for the Bloch-Kato conjecture for the motives and , where is the motif attached to . More precisely, we prove that under certain conditions the -adic valuation of the algebraic part of the symmetric square -function of evaluated at provides a lower bound for the -adic valuation of the order of the Pontryagin...
In this article we study the behavior of inertia groups for modular Galois mod representations and in some cases we give a generalization of Ribet’s lowering the level result (cf. [9]).
Let be two different prime numbers, let be a local non archimedean field of residual characteristic , and let be an algebraic closure of the field of -adic numbers , the ring of integers of , the residual field of . We proved the existence and the unicity of a Langlands local correspondence over for all , compatible with the reduction modulo in [V5], without using and factors of pairs. We conjecture that the Langlands local correspondence over respects congruences modulo between...
We consider the Diophantine equation , where B, D are integers (B ≠ ±2, D ≠ 0) and p is a prime >5. We give Kraus type criteria of nonsolvability for this equation (explicitly, for many B and D) in terms of Galois representations and modular forms. We apply these criteria to numerous equations (with B = 0, 1, 3, 4, 5, 6, specific D’s, and p ∈ (10,10⁶)). In the last section we discuss reductions of the above Diophantine equations to those of signature (p,p,2).