Related papers: Mod pq Galois representations and Serre's conjectu…

In the first part of this paper we try to explain to a general mathematical audience some of the remarkable web of conjectures linking representations of Galois groups with algebraic geometry, complex analysis and discrete subgroups of Lie…

Let $K$ be a $p$-adic local field. In this work we study a special kind of $p$-adic Galois representations of it. These representations are similar to the Galois representations occurred in the exceptional zero conjecture for modular forms.…

We introduce Galois families of modular forms. They are a new kind of family coming from Galois representations of the absolute Galois groups of rational function fields over the rational field. We exhibit some examples and provide an…

Let K be a totally real Galois number field and let A be a set of elliptic curves over K. We give sufficient conditions for the existence of a finite computable set of rational primes P such that for p not in P and E in A, the…

Let $p$ be a prime number and $K$ a finite extension of $\mathbb{Q}_p$. We state conjectures on the smooth representations of $\mathrm{GL}_n(K)$ that occur in spaces of mod $p$ automorphic forms (for compact unitary groups). In particular,…

Let $n>1$, $e\geq 0$ and a prime number $p\geq 2^{n+2+2e}+3$, such that the index of regularity of $p$ is $\leq e$. We show that there are infinitely many irreducible Galois representations $\rho: Gal(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow…

Let $f$ be a newform of weight $k\geq 2$, level $N$ with coefficients in a number field $K$, and $A$ the adjoint motive of the motive $M$ associated to $f$. We carefully discuss the construction of the realisations of $M$ and $A$, as well…

In this paper, we study the fine Selmer groups of two congruent Galois representations over an admissible $p$-adic Lie extension. We show that under appropriate congruence conditions, if the dual fine Selmer group of one is pseudo-null, so…

Given a pair of n-dimensional complex Galois representations over Q, we define their matching density to be the density, if it exists, of the set of places at which the traces of Frobenius of the two Galois representations are equal. We…

Following the natural instinct that when a group operates on a number field then every term in the class number formula should factorize `compatibly' according to the representation theory (both complex and modular) of the group, we are led…

The Galois representation associated to a p-divisible group over a complete noetherian normal local ring with perfect residue field is described in terms of its Dieudonn\'e display. As a corollary we deduce in arbitrary characteristic…

Let $K$ be a finite extension of $\mathbb{Q}_p$. We study the locally $\mathbb{Q}_p$-analytic representations $\pi$ of $\mathrm{GL}_n(K)$ of integral weights that appear in spaces of $p$-adic automorphic representations. We conjecture that…

Let $F$ be a CM field with totally real subfield $F^+$ and let $\pi$ be a $C$-algebraic cuspidal automorphic automorphic representation of $\mathrm{U}(a,b)(\mathbf{A}_{F^+})$ whose archimedean components lie in the (non-degenerate limit of)…

Mazur's principle gives a criterion under which an irreducible mod $\ell$ Galois representation arising from a modular form of level $Np$ (with $p$ prime to $N$) can also arise from a modular form of level $N.$ We prove an analogous result…

We construct and study the moduli of continuous representations of a profinite group with integral $p$-adic coefficients. We present this moduli space over the moduli space of continuous pseudorepresentations and show that this morphism is…

In this paper, we associate Galois representations to globally generic cuspidal automorphic representations on GSp(4), over a totally real field F, which are Steinberg at some finite place. This association is compatible with the local…

We examine which representations of the absolute Galois group of a field of finite characteristic with image over a finite field of the same characteristic may be constructed by the Galois group's action on the division points of an…

We formulate a Serre-type conjecture for n-dimensional Galois representations that are tamely ramified at p. The weights are predicted using a representation-theoretic recipe. For n = 3 some of these weights were not predicted by the…

In this paper we show how to explicitly write down equations of hyperelliptic curves over Q such that for all odd primes l the image of the mod l Galois representation is the general symplectic group. The proof relies on understanding the…

We prove (under certain assumptions) the irreducibility of the limit $\sigma_2$ of a sequence of irreducible essentially self-dual Galois representations $\sigma_k: G_{\mathbf{Q}} \to \mathrm{GL}_4(\overline{\mathbf{Q}}_p)$ (as $k$…