Using class field theory to find function fields with many rational places

Abstract

In this thesis, two algorithms are proposed to find function fields with many rational places. These algorithms rely on the main theorem of class field theory, which tells us that there exists a functorial bijection between finite index subgroups of the idèle class group and finite extensions of a function field. Combining this with knowledge of the splitting behaviour of places in finite extensions gives one algorithm using unramified extensions and a second using ramified extensions. Running the first algorithm over genus 4 hyperelliptic function fields and the second over genus 2 hyperelliptic function fields, both over F_p with prime between 3 and 13, gives 51 results that are improvements to the bounds currently stated on manypoints.org.