Related papers: Constructing Galois representations ramified at on…

It is proved that every two-dimensional residual Galois representation of the absolute Galois group of an arbitrary number field lifts to a characteristic zero $p$-adic representation, if local lifting problems at places above $p$ are…

Let p be a prime and K be a number field. Let rho_{E,p}:G_K \longrightarrow Aut(T_p E)\cong GL_2(Z_p) be the Galois representation given by the Galois action on the p-adic Tate module of an elliptic curve E over K. Serre showed that the…

We prove two "large images" results for the Galois representations attached to a degree $d$ Q-curve $E$ over a quadratic field $K$: if $K$ is arbitrary, we prove maximality of the image for every prime $p >13$ not dividing $d$, provided…

Irreducible representations are the building blocks of general, semisimple Galois representations \rho, and cuspidal representations are the building blocks of automorphic forms \pi of the general linear group. It is expected that when an…

We prove a criterion for the irreducibility of an integral group representation \rho over the fraction field of a noetherian domain R in terms of suitably defined reductions of \rho at prime ideals of R. As applications, we give…

For every finite group $H$ and every finite $H$-module $A$, we determine the subgroup of negligible classes in $H^2(H,A)$, in the sense of Serre, over fields with enough roots of unity. As a consequence, we show that for every odd prime…

In this article, we prove that for a convergent sequence of residually absolutely irreducible representations of the absolute Galois group of a number field $F$ with coefficients in a domain finite over a power series ring over a $p$-adic…

Let X be a normal complex algebraic variety, and p a prime. We show that there exists an integer N=N(X, p) such that: any non-trivial, irreducible representation of the fundamental group of X, which arises from geometry, must be non-trivial…

Let $p$ be prime, and $n,m \in \mathbb{N}$. When $K/F$ is a cyclic extension of degree $p^n$, we determine the $\mathbb{Z}/p^m\mathbb{Z}[\text{Gal}(K/F)]$-module structure of $K^\times/K^{\times p^m}$. With at most one exception, each…

Let $p>2$ be a prime. We give examples of smooth absolutely irreducible representations of $\mathrm{GL}_2(\mathbb{Q}_{p^3})$ over $\mathbb{F}_{p^3}$ which are not admissible.

If a $p$-adic Galois representation $\rho_{f,\nu}:\Gamma_{\mathbb Q} \to \GL_2(E_{f,\nu})$ attached to some eigenform $f$ is residually reducible it will have 2 non-isomorphic reductions, which have the same semi-simplification. In this…

Let $G$ be a finite group and let $ram^{t}(G)$ denote the minimal positive integer $n$ such that $G$ can be realized as the Galois group of a tamely ramified extension of $\mathbb{Q}$ ramified only at $n$ finite primes. Let $d(G)$ denote…

For a rational prime $p\neq 2$, we compute the sequence of ramification groups of a Galois, radical and finite extension $L/F$ where $F/\mathbb{Q}_p$ is an unramified finite extension. First, we compute it in the case where the exponent of…

We construct motivic $\ell$-adic representations of $\GQ$ into exceptional groups of type $E_7,E_8$ and $G_2$ whose image is Zariski dense. This answers a question of Serre. The construction is uniform for these groups and uses the…

Let $p$ be a prime number and $F$ a totally real number field unramified at places above $p$. Let $\bar{r}:\operatorname{Gal}(\bar F/F)\rightarrow\operatorname{GL}_2(\bar{\mathbb{F}_p})$ be a modular Galois representation which satisfies…

To a hyperbolic smooth curve defined over a number-field one naturally associates an "anabelian" representation of the absolute Galois group of the base field landing in outer automorphism group of the algebraic fundamental group. In this…

We study the rigid generic fiber $\mathcal{X}^\square_{\overline\rho}$ of the framed deformation space of the trivial representation $\overline\rho: G_K \to \text{GL}_n(k)$ where $k$ is a finite field of characteristic $p>0$ and $G_K$ is…

Let $p$ be an odd prime. Let $F$ be a non-archimedean local field of residue characteristic $p$, and let $\mathbb{F}_q$ be its residue field. Let $\mathcal{H}^{(1)}_{\mathbb{F}_q}$ be the pro-$p$-Iwahori-Hecke algebra of the $p$-adic group…

We show how the output of the algorithm to compute modular Galois representations described in our previous article can be certified. We have used this process to compute certified tables of such Galois representations obtained thanks to an…

We study the possible weights of an irreducible 2-dimensional modular mod p representation of the absolute Galois group of F, where F is a totally real field which is totally ramified at p, and the representation is tamely ramified at the…