THE BRAUER-MANIN OBSTRUCTION ON FAMILIES OF HYPERELLIPTIC CURVES

Abstract

OF THE DISSERTATION The Brauer-Manin Obstruction on Families of Hyperelliptic Curves by Thom Tyrrell Dissertation Director: Jerrold Tunnell In [19], Manin introduced a way to explain the failure of the Hasse principle for algebraic varieties over a number field. For curves, the problem of whether all such failures can be explained by this method is open. In this thesis, we construct unramified quaternion algebras that obstruct the existence of rational points on families of curves admitting maps to elliptic curves. In Chapter 1, we introduce the Brauer group of a projective variety. For curves, we relate the Brauer group to torsors over the Jacobian of the curve, and establish some properties of the Brauer group over local fields that will simplify later computations. In Chapter 2, we define the Brauer-Manin obstruction and show that it explains all failures of the Hasse principle for genus 1 curves and curves without a degree 1 rational divisor. After reviewing a topological and more computational characterization of the obstruction, we note that it suffices to consider curves that admit maps to positive rank abelian varieties. Having established the obstruction, we introduce the motivating example for this thesis a construction by D.Quan [21] of a genus 11 hyperelliptic curve and an unramified quaternion algebra that obstructs the existence of rational points on it.