Period -Index problem for hyperelliptic curves

TL;DR

This paper investigates the relationship between the period and index of elements in the 2-torsion of the Tate-Shafarevich group of the Brauer group of genus 2 hyperelliptic curves over number fields, establishing their equality and exploring related moduli space isomorphisms.

Contribution

It proves the equality of period and index for 2-torsion elements in the Tate-Shafarevich group of certain hyperelliptic curves and establishes a new isomorphism of moduli spaces with Grassmannians.

Findings

Period and index coincide for 2-torsion elements in Sha(Br(C)).

An isomorphism between moduli space of rank 2 bundles and Grassmannian is established.

A weak Hasse principle for intersections of two quadrics in projective 5-space is derived.

Abstract

Let $C$ be a smooth projective curve of genus 2 over a number field $k$ with a rational point. We prove that the index and exponent coincide for elements in the 2-torsion of $\Sha(Br(C))$. In the appendix, an isomorphism of the moduli space of rank 2 stable vector bundles with odd determinant on a smooth projective hyperelliptic curve $C$ of genus $g$ with a rational point over any field of characteristic not two with the Grassmannian of $(g-1)$-dimensional linear subspaces in the base locus of a certain pencil of quadrics is established, making a result of (\cite{De-Ra}) rational. We establish a twisted version of this isomorphism and we derive as a consequence a weak Hasse principle for the smooth intersection $X$ of two quadrics in ${\mathbb P}^5$ over a number field: if $X$ contains a line locally, then $X$ has a $k$-rational point.