The arithmetic of uniquely trigonal genus 4 curves

Number fields, which are finite field extensions of the field of rational numbers, are fundamental objects of study in number theory. One of the most important invariants of a number field is the class group, which measures how close the ring of integers of that number field is to being a unique factorization domain. Despite their importance, class groups remain mysterious objects in number theory. This talk will demonstrate how the titular family of curves can be used to construct cubic number fields with interesting class groups.

Background: The talk is designed for a general level mathematics colloquium. Some algebraic number theory or algebraic geometry is useful, but not necessary.