Related papers: Realizing Galois representations in abelian variet…

The aim of this paper is to extend our old results about Galois action on the torsion points of abelian varieties to the case of (finitely generated) fields of characteristic 2.

In his previous paper (Math. Res. Letters 7(2000), 123--132) the author proved that in characteristic zero the jacobian $J(C)$ of a hyperelliptic curve $C: y^2=f(x)$ has only trivial endomorphisms over an algebraic closure of the ground…

We establish non-unirational versions of Hilbert Irreducibility for all Hilbert modular surfaces which are of K3 type. As an application we prove new instances of the regular Inverse Galois Problem for the simple groups…

We consider the groups G which arise from real semisimple Jordan algebras via the Tits-Koecher-Kantor construction. Such a G is characterized by the fact that it admits a parabolic subgroup P=LN which is conjugate to its opposite, and for…

Given an elliptic curve $E$ over a number field $K$, the $\ell$-torsion points $E[\ell]$ of $E$ define a Galois representation $\gal(\bar{K}/K) \to \gl_2(\ff_\ell)$. A famous theorem of Serre states that as long as $E$ has no Complex…

In this work we generalise the main result of arXiv:1812.05651 to the family of hyperelliptic curves with potentially good reduction over a $p$-adic field which have degree $p$ and the largest possible image of inertia under the $\ell$-adic…

Fix a prime $p > 2$. Let $\rho : \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \text{GL}_2(\mathbb{I})$ be the Galois representation coming from a non-CM irreducible component $\mathbb{I}$ of Hida's $p$-ordinary Hecke algebra. Assume the…

An algebraic extension of the rational numbers is said to have the $\textit{Bogomolov property}$ (B) if the absolute logarithmic Weil height of its non-torsion elements is uniformly bounded from below. Given a continuous representation…

For a genus $2$ curve $C$ over $\mathbb{Q}$ whose Jacobian $A$ admits only trivial geometric endomorphisms, Serre's open image theorem for abelian surfaces asserts that there are only finitely many primes $\ell$ for which the Galois action…

Let $k$ be a subfield of $\mathbb{C}$ which contains all $2$-power roots of unity, and let $K = k(\alpha_{1}, \alpha_{2}, ... , \alpha_{2g + 1})$, where the $\alpha_{i}$'s are independent and transcendental over $k$, and $g$ is a positive…

Let $K/F$ be a finite Galois extension of number fields. It is well known that the Tchebotarev density theorem implies that an irreducible, finitely ramified $p$-adic representation $\rho$ of the absolute Galois group of $K$ is determined…

Let $K$ be a number field of degree $d$ so that $K/\mathbb Q$ is a Galois extension. The {\it normal basis theorem} states that $K$ has a $\mathbb Q$-basis consisting of algebraic conjugates, in fact $K$ contains infinitely many such bases.…

Given a number field $K$, a finite abelian group $G$ and finitely many elements $\alpha_1,\ldots,\alpha_t\in K$, we construct abelian extensions $L/K$ with Galois group $G$ that realise all of the elements $\alpha_1,\ldots,\alpha_t$ as…

Let $p$ be a prime number, $n$ an integer $\geq 2$ and $\rho$ an $n$-dimensional automorphic $p$-adic Galois representation (for a compact unitary group) such that $r:=\rho\vert_{\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)}$ is…

Suppose \( \rho_1 \) and \( \rho_2 \) are two pure Galois representations of the absolute Galois group of a number field $K$ of weights \( k_1 \) and \( k_2 \) respectively, having equal normalized Frobenius traces \( Tr(\rho_1(\sigma_v))…

The aim of Bogomolov's programme is to prove birational anabelian conjectures for function fields $K|k$ of varieties of dimension $\geq 2$ over algebraically closed fields. The present article is concerned with the 1-dimensional case. While…

The main purpose of this paper is to describe the abelian part $\mathcal G^{ab}_{K}$ of the absolute Galois group of a global function field $K$ as pro-finite group. We will show that the characteristic $p$ of $K$ and the non $p$-part of…

We use Galois closures of finite rational maps between complex projective varieties to introduce a new method for producing varieties such that the holomorphic part of the cup product map has non-trivial kernel. We then apply our result to…

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)…

We construct a Langlands parameterization of supercuspidal representations of $G_2$ over a $p$-adic field. More precisely, for any finite extension $K / \QQ_p$ we will construct a bijection \[ \CL_g : \CA^0_g(G_2,K) \rightarrow \CG^0(G_2,K)…