Rational points on abelian varieties over function fields and Prym varieties

In this paper, using a generalization of the notion of Prym variety for covers of projective varieties, we prove a structure theorem for the Mordell–Weil group of abelian varieties over function fields that are twists of abelian varieties by Galois covers of smooth projective varieties. In particular, the results we obtain contribute to the construction of Jacobians of high rank.

Semi-abelian Spectral Data for Singular Fibres of the 𝖲𝖫(2,ℂ)-Hitchin System

We describe spectral data for singular fibres of the $\textsf{SL}(2,{\mathbb{C}})$-Hitchin fibration with irreducible and reduced spectral curve. Using Hecke transformations, we give a stratification of these singular spaces by fibre bundles over Prym varieties. By analysing the parameter spaces of Hecke transformations, this describes the singular Hitchin fibres as compactifications of abelian group bundles over abelian torsors. We prove that a large class of singular fibres are themselves fibre bundles over Prym varieties. As applications, we study irreducible components of singular Hitchin fibres and give a description of $\textsf{SL}(2,{\mathbb{R}})$-Higgs bundles in terms of these semi-abelian spectral data.

Irrationality of generic cubic threefold via Weil's conjectures

An arithmetic method of proving the irrationality of smooth projective 3-folds is described, using reduction modulo [Formula: see text]. It is illustrated by an application to a cubic threefold, for which the hypothesis that its intermediate Jacobian is isomorphic to the Jacobian of a curve is contradicted by reducing modulo 3 and counting points over appropriate extensions of [Formula: see text]. As a spin-off, it is shown that the 5-dimensional Prym varieties arising as intermediate Jacobians of certain cubic 3-folds have the maximal number of points over [Formula: see text] which attains Perret's and Weil's upper bounds.