On the Arithmetic of Hyperelliptic Curves

Abstract

My research involves answering various number-theoretic questions involving hyperelliptic curves. A hyperelliptic curve is a generalization of elliptic curves to curves of higher genus but which still have explicit equations. The first part of this thesis involves examining moduli of hyperelliptic curves and in particular, compare their field of moduli with possible fields of definition of the curve. For even genus g, a general hyperelliptic curve of genus g cannot be defined over its field of moduli. Meanwhile, in odd genus, this can be done for curves which admit only two automorphisms, but the curve constructed may not be a hyperelliptic curve in the familiar sense. The second part involves the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point. The 2-Selmer group is a subgroup of the Galois cohomology group H^1 (K, J[2]), and J[2] is preserved by quadratic twists, so one may consider how the 2-Selmer rank varies over quadratic twists. I show that starting with any hyperelliptic curve with a rational Weierstrass point, the 2-Selmer rank is unbounded over quadratic twists.