Related papers: Abelian varieties are not quotients of low-dimensi…

We prove the existence of an Abelian variety $A$ of dimension $g$ over $\Qa$ which is not isogenous to any Jacobian, subject to the necessary condition $g>3$. Recently, C.Chai and F.Oort gave such a proof assuming the Andr\'e-Oort…

Let $A$ be an abelian variety, and $G \subset Aut(A)$ a finite group acting freely in codimension two. We discuss whether the singular quotient $A/G$ admits a resolution that is a Calabi-Yau manifold. While Oguiso constructed two examples…

We prove surjectivity criteria for $p$-adic representations and we apply them to abelian varieties over number fields. In particular, we provide examples of Jacobians over $\dbQ$ of dimension $d\in\{1,2,3\}$ whose 2-adic representations…

Given a polynomial $f\in\mathbb{C}[x]$, we consider the family of superelliptic curves $y^d=f(x)$ and their Jacobians $J_d$ for varying integers $d$. We show that for any integer $g$ the number of abelian varieties up to isogeny of…

We study the rationality properties of the moduli space $\mathcal{A}_g$ of principally polarised abelian $g$-folds over $\mathbb{Q}$ and apply the results to arithmetic questions. In particular we show that any principally polarised abelian…

We show that polarisations of type (1,...,1,2g+2) on g-dimensional abelian varieties are $\it{never}$ very ample, if $g\geq 3$. This disproves a conjecture of Debarre, Hulek and Spandaw. We also give a criterion for non-embeddings of…

We prove the existence of abelian varieties over $\overline{\mathbb Q(t)}$ with no power isogenous to a Jacobian. Moreover, given a positive integer $N$, we prove the existence of abelian varieties over $\overline{\mathbb Q(t)}$ with…

Inspired by experimental data, this paper investigates which isogeny classes of abelian varieties defined over a finite field of odd characteristic contain the Jacobian of a hyperelliptic curve. We provide a necessary condition by…

The characteristic polynomials of abelian varieties over the finite field $\mathbb{F}_q$ with $q=p^n$ elements have a lot of arithmetic and geometric information. They have been explicitly described for abelian varieties up to dimension 4,…

We prove the existence of abelian varieties not isogenous to Jacobians over characterstic $p$ function fields. Our methods involve studying the action of degree $p$ Hecke operators on hypersymmetric points, as well as their effect on the…

We define and study Jacobians of Hodge structures with weight greater than 1. Jacobians of weight 2 naturally come up in the context of the Brauer group and the Tate conjecture. They were previously studied in a special case by Beauville in…

For a curve of genus at least four which is either very general or very general hyperelliptic, we classify all ways in which a power of its Jacobian can be isogenous to a product of Jacobians of curves. As an application, we show that, for…

We construct two abelian varieties over $\mathbb{Q}$ which are not isomorphic, but have isomorphic Mordell--Weil groups over every number field, isomorphic Tate modules and equal values for several other invariants.

In this article, we show that for any non-isotrivial family of abelian varieties over a rational base with big monodromy, those members that have adelic Galois representation with image as large as possible form a density-$1$ subset. Our…

We prove by means of the study of the infinitesimal variation of Hodge structure and a generalization of the classical Babbage-Enriques-Petri theorem that the Jacobian variety of a generic element of a $k$ codimensional subvariety of…

Let $K$ be a number field, let $g \geq 1$ be an integer and let $f(x) = (x - a_1) \cdots (x - a_{2g + 1}) \in O_K[x]$ be a polynomial that splits into $2g + 1$ distinct linear factors. Write $C$ for the hyperelliptic curve given by $C: y^2…

In this article we construct for each integer $n\ge 2$ an abelian variety $A$ of dimension $n$ defined over a number field for which there exists a symmetric slope sequence of length $2n$ that does not appear as the slope sequence of…

Let l be a prime. We show that there do not exist any non-zero semi-stable abelian varieties over Q with good reduction outside l if and only if l=2, 3, 5, 7 or 13. We show that any semi-stable abelian variety over Q with good reduction…

We show how for every integer n one can explicitly construct n distinct plane quartics and one hyperelliptic curve over the complex numbers all of whose Jacobians are isomorphic to one another as abelian varieties without polarization. When…

We study the set of isomorphism classes of principal polarizations on abelian varieties of GL2-type. As applications of our results, we construct examples of curves C, C'/\Q of genus two which are nonisomorphic over \bar \Q and share…