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 $p\le 2593$, a lower bound for the number of isomorphism classes of Galois representation of $\mathbf{Q}$ on a two–dimensional vector space over ${\overline{\mathbf{F}}}_p$ which are irreducible, odd, and unramified outside $p$.

Let $k$ be a positive integer divisible by 4, $p\>k$ a prime, $f$ an elliptic cuspidal eigenform (ordinary at $p$) of weight $k-1$, level 4 and non-trivial character. In this paper we provide evidence for the Bloch-Kato conjecture for the motives ${\text{ad}}^{0}M(-1)$ and ${\text{ad}}^{0}M\left(2\right)$, where $M$ is the motif attached to $f$. More precisely, we prove that under certain conditions the $p$-adic valuation of the algebraic part of the symmetric square $L$-function of $f$ evaluated at $k$ provides a lower bound for the $p$-adic valuation of the order of the Pontryagin...

In this article we study the behavior of inertia groups for modular Galois mod ${\ell }^n$ representations and in some cases we give a generalization of Ribet’s lowering the level result (cf. [9]).

Let $\ell \ne p$ be two different prime numbers, let $F$ be a local non archimedean field of residual characteristic $p$, and let ${\overline{𝐐}}_{\ell },{\overline{𝐙}}_{\ell },{\overline{𝐅}}_{\ell }$ be an algebraic closure of the field of $\ell$-adic numbers ${𝐐}_{\ell }$, the ring of integers of ${\overline{𝐐}}_{\ell }$, the residual field of ${\overline{𝐙}}_{\ell }$. We proved the existence and the unicity of a Langlands local correspondence over ${\overline{𝐅}}_{\ell }$ for all $n\ge 2$, compatible with the reduction modulo $\ell$ in [V5], without using $L$ and $ϵ$factors of pairs. We conjecture that the Langlands local correspondence over ${\overline{𝐐}}_{\ell }$ respects congruences modulo $\ell$ between...

We consider the Diophantine equation $(x+y)(x²+Bxy+y²)=D{z}^p$, 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).

In this paper we generalize the deformation theory of representations of a profinite group developed by Schlessinger and Mazur to deformations of objects of the derived category of bounded complexes of pseudocompact modules for such a group. We show that such objects have versal deformations under certain natural conditions, and we find a sufficient condition for these versal deformations to be universal. Moreover, we consider applications to deforming Galois cohomology classes and the étale hypercohomology...

An effective construction of homogeneous linear differential equations of order 2 with Galois group $2A_4,2S_4$ or $2A_5$ is presented.

In this paper, we develop the Euler system theory for Galois deformations. By applying this theory to the Beilinson-Kato Euler system for Hida’s nearly ordinary modular deformations, we prove one of the inequalities predicted by the two-variable Iwasawa main conjecture. Our method of the proof of the Euler system theory is based on non-arithmetic specializations. This gives a new simplified proof of the inequality between the characteristic ideal of the Selmer group of a Galois deformation and the...

Dans cet article nous présentons la théorie des équations différentielles $p$-adiques et ses applications concernant le théorème de finitude de la cohomologie $p$-adique d’une variété affine et le théorème de la monodromie $p$-adique des représentations galoisiennes locales.