Related papers: A relative Shafarevich theorem

For every odd prime number p, we give examples of non-constant smooth families of genus 2 curves over fields of characteristic p which have pro-Galois (pro-\'etale) covers of infinite degree with geometrically connected fibers. The…

Let U be a universal covering of a connected nonsingular projective variety X with large and residually finite fundamental group. We construct metrics on U and provide another version of the uniformization theorem, namely: if the…

We show that the theory of Galois actions of a torsion Abelian group $A$ is companionable if and only if for each prime $p$, the $p$-primary part of $A$ is either finite or it coincides with the Pr\"{u}fer $p$-group. We also provide a…

Let E/K be an elliptic curve defined over a number field, and let p be a prime number such that E(K) has full p-torsion. We show that the order of the p-part of the Shafarevich-Tate group of E/L is unbounded as L varies over degree p…

Let $A$ be an absolutely simple abelian variety without (potential) complex multiplication, defined over the number field $K$. Suppose that either $\dim A=2$ or $A$ is of $\operatorname{GL}_2$-type: we give an explicit bound $\ell_0(A,K)$…

Given a Galois cover of curves over $\mathbb{F}_p$, we relate the $p$-adic valuation of epsilon constants appearing in functional equations of Artin L-functions to an equivariant Euler characteristic. Our main theorem generalises a result…

This paper computes the Galois group of the Galois cover of the composition of an \'etale double cover of a cyclic $p$-gonal cover for any prime $p$. Moreover a relation between some of its Prym varieties and the Jacobian of a subcover is…

Fix an abelian variety $A$ of dimension $g\geq 1$ defined over a number field $K$. For each prime $\ell$, the Galois action on the $\ell$-power torsion points of $A$ induces a representation $\rho_{A,\ell}\colon Gal_K \to…

Let $Z \to X$ be a finite branched Galois cover of normal projective geometrically integral varieties of dimension $d \geq 2$ over a perfect field $k$. For such a cover, we prove a Chebotarev-type density result describing the decomposition…

In this paper we generalize results of P. Le Duff to genus n hyperelliptic curves. More precisely, let C/Q be a hyperelliptic genus n curve and let J(C) be the associated Jacobian variety. Assume that there exists a prime p such that J(C)…

In this paper, we give a necessary and sufficient condition for the finiteness of Galois cohomology of unipotent groups over local fields of positive characteristic

In this paper we present a complete proof of I.R.Safarevic's famous theorem that every finite solvable group occurs as a Galois group over Q.

Let ${\mathcal L}/{\mathcal K}$ be a finite Galois extension and let $X$ be an affine algebraic variety defined over ${\mathcal L}$. Weil's Galois descent theorem provides necessary and sufficient conditions for $X$ to be definable over…

We revisit Kolchin's results on definability of differential Galois groups of strongly normal extensions, in the case where the field of constants is not necessarily algebraically closed. In certain classes of differential topological…

Given a Galois cover of curves $f$ over a field of characteristic $p$, the lifting problem asks whether there exists a Galois cover over a complete mixed characteristic discrete valuation ring whose reduction is $f$. In this paper, we…

In this paper, we consider infinite Galois extensions of number fields and study the relation between their local degrees and the structure of their Galois groups. It is known that, if $K$ is a number field and $L/K$ is an infinite Galois…

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 $E$ be an optimal elliptic curve over $\Q$ of prime conductor $N$. We show that if for an odd prime $p$, the mod $p$ representation associated to $E$ is reducible (in particular, if $p$ divides the order of the torsion subgroup of…

We define a proper moduli stack for degree $p$ covers $f:Y \to \cX$ where $\cX$ is a twisted stable curve in the sense of [5] and [4], and $Y$ is a stable curve which via $f$ is a torsor over $\cX$ under a finite flat group scheme $\cG \to…

We make explicit certain results around the Galois correspondence in the context of definable automorphism groups, and point out the relation to some recent papers dealing with the Galois theory of algebraic differential equations when the…