Related papers: Minimal Subgroups of ${\rm GL}_2(\mathbb{Z}_{S})$

In 1972, Serre showed that the adelic Galois representation associated to a non-CM elliptic curve over a number field has open image in GL_2(\hat{Z}). In Greicius' thesis, he develops necessary and sufficient criteria for determining when…

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $G$ be $Z/\ell Z$ semi-direct product $Z/pZ$ where $\ell$ is a prime distinct from $p$. In this paper, we study Galois covers $\psi:Z \to P^1_k$ ramified only over…

We prove an explicit surjectivity result for products of non-isotrivial, non-isogenous elliptic curves over a function field of arbitrary characteristic. This is by way of an isogeny degree bound in this setting, generated from bounds for…

In this note we give a self-contained proof of the following classification (up to conjugation) of subgroups of the general symplectic group of dimension n over a finite field of characteristic l, for l at least 5, which can be derived from…

Although it is not known which groups can appear as torsion groups of elliptic curves over cubic number fields, it is known which groups can appear for infinitely many non-isomorphic curves. We denote the set of these groups as $S$. In this…

Let $E/\mathbb{Q}$ be an elliptic curve, let $\overline{\mathbb{Q}}$ be a fixed algebraic closure of $\mathbb{Q}$, and let $G_{\mathbb{Q}}=\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ be the absolute Galois group of $\mathbb{Q}$. The…

In his previous papers (Math. Res. Letters 7 (2000), 123--13; Math. Res. Letters 8 (2001), 429--435; Moscow Math. J. 2 (2002), issue 2, 403-431) the author proved that in characteristic $\ne 2$ the jacobian $J(C)$ of a hyperelliptic curve…

This article advances the results of Duke on the average surjectivity of Galois representations for elliptic curves to the context of Drinfeld modules over function fields. Let $F$ be the rational function field over a finite field. I…

In this paper, we prove a minimal modularity lifting theorem for Galois representations (conjecturally) associated to Siegel modular forms of genus two which are holomorphic limits of discrete series at infinity.

In this paper, we study the Galois representations attached to products of Drinfeld modules. As an analogue of Serre's classical result on the images of Galois representations associated with products of elliptic curves, we prove that for…

The l-adic parabolic cohomology groups attached to noncongruence subgroups of SL_2(Z) are finite-dimensional representations of Gal(Qbar/F) for some number field F. We exhibit examples (with F=Q) giving rise to Galois representations whose…

Let $E$ be an elliptic curve without complex multiplication defined over the rationals. The purpose of this article is to define a positive integer $A(E)$, that we call the {\it Serre's constant associated to $E$}, that gives necessary…

Serre's uniformity question asks whether there exists a bound $N>0$ such that, for every non-CM elliptic curve $E$ over $\mathbb{Q}$ and every prime $p>N$, the residual Galois representation…

Let k be a global field, p an odd prime number different from char(k) and S, T disjoint, finite sets of primes of k. Let G_S^T(k)(p)=Gal(k_S^T(p)|k) be the Galois group of the maximal p-extension of k which is unramified outside S and…

If $X'= ({\mathbb B} / \Gamma)'$ is a torsion free toroidal compactification of a discrete ball quotient $X_o={\mathbb B} / \Gamma$ and $\xi : (X', T = X'\setminus X_o) \rightarrow (X, D = \xi (T))$ is the blow-down of the $(-1)$-curves to…

A problem of current interest, also motivated by applications to Coding theory, is to find explicit equations for \textit{maximal} curves, that are projective, geometrically irreducible, non-singular curves defined over a finite field…

Let G be a connected and reductive algebraic group over an algebraically closed field of characteristic p > 0. An interesting class of representations of G consists of those G-modules having a good filtration -- i.e. a filtration whose…

Let $K$ be a number field and $E_1, \ldots, E_n$ be elliptic curves over $K$, pairwise non-isogenous over $\overline{K}$ and without complex multiplication over $\overline{K}$. We study the image of the adelic representation of the absolute…

In the 80's Aschbacher classified the maximal subgroups of almost all of the finite almost simple classical groups. Essentially, this classification divide these subgroups into two types. The first of these consist roughly of subgroups that…

Let $X$ be a minimal cubic surface over a finite field $\mathbb{F}_q$. The image $\Gamma$ of the Galois group $\operatorname{Gal}(\overline{\mathbb{F}}_q / \mathbb{F}_q)$ in the group $\operatorname{Aut}(\operatorname{Pic}(\overline{X}))$…