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).