The Brauer-Manin obstruction on families of hyperelliptic curves

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. ii The next two chapters may be roughly described as the "global" and "local" compo- nents of this thesis. In Chapter 3, we review the geometry of split Jacobians and ruled surfaces. We then use recent work of Doerksen and Bruin [2] on the characterization of (4,4)-split Jacobians to analyze the ramification of the algebra introduced by Quan. In Chapter 4, we apply the results of the previous chapter to construct quaternion algebras on a family of hyperelliptic curves of genus 5. With the local results of Chap- ter 1, we give a new analysis of Quan’s example, and show these quaternion algebras obstruct the existence of rational points in the family. In Chapter 5, we briefly describe a conjecture of Poonen concerning the topological characterization of the Brauer-Manin obstruction given in the second chapter. With code executed in sage, we provide additional computational evidence for the conjecture that all failures of the Hasse principle for curves are explained by the obstruction.