Related papers: The Large Sieve and Galois Representations

We prove a generalization of a conjecture of C. Marion on generation properties of finite groups of Lie type, by considering geometric properties of an appropriate representation variety and associated tangent spaces.

Following Isaacs (see [Isa08, p. 94]), we call a normal subgroup N of a finite group G large, if $C_G(N) \leq N$, so that N has bounded index in G. Our principal aim here is to establish some general results for systematically producing…

Fix a positive integer $g$ and rational prime $p$. We prove the existence of a genus $g$ curve $C/\mathbb{Q}$ such that the mod $p$ representation of its Jacobian is tame by imposing conditions on the endomorphism ring. As an application,…

Let $A$ be a $g$-dimensional abelian variety over $\mathbb{Q}$ whose adelic Galois representation has open image in $\text{GSp}_{2g} \widehat{\mathbb{Z}}$. We investigate the endomorphism algebras $\text{End}(A_p) \otimes \mathbb{Q} =…

We formulate a notion of purity for $p$-adic big Galois representations and pseudorepresentations of Weil groups of $\ell$-adic number fields for $\ell\neq p$. This is obtained by showing that all powers of the monodromy of any big Galois…

The paper is concerned with the following version of Hilbert's irreducibility theorem: if $\pi: X \to Y$ is a Galois $G$-covering of varieties over a number field $k$ and $H \subset G$ is a subgroup, then for all sufficiently large and…

We give a complete description of the arboreal Galois representation of a certain postcritically finite cubic polynomial over a large class of number fields and for a large class of basepoints. This is the first such example that is not…

Let $E/\mathbb{Q}$ be an elliptic curve without complex multiplication. By Serre's open image theorem, the mod $\ell$ Galois representation $\overline{\rho}_{E, \ell}$ of $E$ is surjective for each prime number $\ell$ that is sufficiently…

To define the notion of a generic property of finite dimensional 2-step nilpotent Lie algebras we use standard correspondence between such Lie algebras and points of an appropriate algebraic variety, where a negligible set is one contained…

A generalization of Serre's Conjecture asserts that if $F$ is a totally real field, then certain characteristic $p$ representations of Galois groups over $F$ arise from Hilbert modular forms. Moreover it predicts the set of weights of such…

In analogy with epsilon-biased sets over Z_2^n, we construct explicit epsilon-biased sets over nonabelian finite groups G. That is, we find sets S subset G such that | Exp_{x in S} rho(x)| <= epsilon for any nontrivial irreducible…

The existence of rational points on Kummer varieties associated to 2-coverings of abelian varieties over number fields can sometimes be proved through the variation of the Selmer group in the family of quadratic twists of the underlying…

We prove a local Gan-Gross-Prasad conjecture on predicting the branching law for the non-tempered representations of general linear groups in the case of non-Archimedean field. We also generalize to Bessel and Fourier-Jacobi models and…

In this paper we extend a result for representations of the Additive group $G_a$ given in [3] to the Heisenberg group $H_1$. Namely, if $p$ is greater than 2d then all $d$-dimensional characteristic $p$ representations for $H_1$ can be…

We prove a general large sieve statement in the context of random walks on subgraphs of a given graph. This can be seen as a generalization of previously known results where one performs a random walk on a group enjoying a strong spectral…

From the generalized Riemann hypothesis for motivic L-functions, we derive an effective version of the Sato-Tate conjecture for an abelian variety A defined over a number field k with connected Sato-Tate group. By effective we mean that we…

We extend the Brauer-Siegel theorem to new families of number fields, both in the classical setting of asymptotically bad families and in the more general framework due to Tsfasman and Vl\u{a}du\c{t} of asymptotically exact families. We…

An algebraic technique is presented that does not use results of model theory and makes it possible to construct a general Galois theory of arbitrary nonlinear systems of partial differential equations. The algebraic technique is based on…

We are considering iterative derivations on the function field L of abelian schemes in positive characteristic p>0, and give conditions when the torsion group schemes of this abelian scheme occur as ID-automorphism groups, i.e. are the…

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…