Related papers: Constructing abelian varieties from rank 2 Galois …

We give a categorical description of all abelian varieties with commutative endomorphism ring over a finite field with $q=p^a$ elements in a fixed isogeny class in terms of pairs consisting of a fractional $\mathbb Z[\pi,q/\pi]$-ideal and a…

The main result of this article is the fact that the currents defined by Levin give a description of the polylogarithm of an abelian scheme at the topological level. This result was a conjecture of Levin. This provides a method to explicit…

Let A be a supersingular abelian variety over a finite field k. We give an approximate description of the structure of the group A(k) of rational points of A over k in terms of the characteristic polynomial f of the Frobenius endomorphism…

In this paper we prove the Geyer-Jarden conjecture on the torsion part of the Mordell-Weil group for a large class of abelian varieties defined over finitely generated fields of arbitrary characteristic. The class consists of all abelian…

We study the family of algebraic curves of genus $\geq 1$ defined by the affine equations $y^s=ax^r+b$ over a number field $k$, where $r \geq 2$ and $s\geq 2$ are fixed integers. Assuming the strong version of Lang's conjecture on varieties…

Let $X_{1}$ be a projective, smooth and geometrically connected curve over $\mathbb{F}_{q}$ with $q=p^{n}$ elements where $p$ is a prime number, and let $X$ be its base change to an algebraic closure of $\mathbb{F}_{q}$. We give a formula…

Let $K$ be a field finitely generated over ${\Q}$, and $A$ an Abelian variety defined over $K$. Then by the Mordell-Weil Theorem, the set of rational points $A(K)$ is a finitely-generated Abelian group. In this paper, assuming Tate's…

Let $[X,\lambda]$ be a principally polarized abelian variety over a finite field with commutative endomorphism ring; further suppose that either $X$ is ordinary or the field is prime. Motivated by an equidistribution heuristic, we introduce…

Let $\{\rho_\ell\}_\ell$ be the system of $\ell$-adic representations arising from the $i$th $\ell$-adic cohomology of a complete smooth variety $X$ defined over a number field $K$. Let $\Gamma_\ell$ and $\mathbf{G}_\ell$ be respectively…

Given a natural number n and a number field K, we show the existence of an integer \ell_0 such that for any prime number \ell\geq \ell_0, there exists a finite extension F/K, unramified in all places above \ell, together with a principally…

We study Galois representations attached to nonsimple abelian varieties over finitely generated fields of arbitrary characteristic. We give sufficient conditions for such representations to decompose as a product, and apply them to prove…

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…

It follows from the Grothendieck-Ogg-Shafarevich formula that the rank of an abelian variety (with trivial trace) defined over the function field of a curve is bounded by a quantity which depends on the genus of the base curve and on bad…

Let A_2 be the moduli stack of principally polarized abelian surfaces and V a smooth l-adic sheaf on A_2 associated to an irreducible rational finite dimensional representation of Sp(4). We give an explicit expression for the cohomology of…

Motivated by our arithmetic applications, we required some tools that might be of independent interest. Let $\mathcal E$ be an absolutely irreducible group scheme of rank $p^4$ over $\mathbb Z_p$. We provide a complete description of the…

Let $p$ be a prime, let $K$ be a discretely valued extension of $\mathbb{Q}_p$, and let $A_{K}$ be an abelian $K$-variety with semistable reduction. Extending work by Kim and Marshall from the case where $p>2$ and $K/\mathbb{Q}_p$ is…

Let $\rho$ be a finite-dimensional faithful representation of a semisimple algebraic group $G$. By means of a deformation argument, we show that there exists a family of Abelian varieties over a smooth and projective curve over the…

An abelian variety $A/K$ is heavenly at $\ell$ if the extension $K(A[\ell^\infty])/K(\mu_{\ell^{\infty}}\!)$ is both pro-$\ell$ and unramified away from $\ell$. It is known that for a fixed quadratic field $K$, the number of $K$-isomorphism…

An abelian variety over a number field is called L-abelian variety if, for any element of the absolute Galois group of a number field L, the conjugated abelian variety is isogenous to the given one by means of an isogeny that preserves the…

In this paper, we prove that if the Frobenius traces agree at all but finitely many places, then two $l$-adic Galois representations, associated to rank-$2$ non-CM Drinfeld modules of generic characteristic, are isomorphic. As a…