Aspects of the Arithmetic of Uniquely Trigonal Genus Four Curves: Arithmetic Invariant Theory and Class Groups of Cubic Number Fields

In this thesis, we study the family of uniquely trigonal genus 4 curves via their connection to del Pezzo surfaces of degree 1. We consider two different aspects of these curves. First, we show how to construct any uniquely trigonal genus 4 curve whose Jacobian variety has fully rational 2-torsion and whose trigonal morphism has a prescribed totally ramified fibre. Using this construction, we find an infinite family of cubic number fields whose class group has 2-rank at least 8. We also consider genus 4 curves that are superelliptic of degree 3, and prove sharp results on the size of the 2-torsion subgroup of their Jacobian varieties over the rationals. Our second result is a contribution to arithmetic invariant theory. We consider the moduli space of uniquely trigonal genus 4 curves whose trigonal morphism has a marked ramification point, originally studied by Coble, from a modern perspective.We then show under a technical hypothesis how to construct an assignment from the rational points on the Jacobian variety of such a curve to a fixed orbit space that is independent of the curve. The assignment is compatible with base extensions of the field, and over an algebraically closed field our assignment reduces to the assignment of a marked (in the aforementioned sense) uniquely trigonal genus 4 curve to its moduli point. The orbit space is constructed from the split algebraic group of type E8.