Erland Arctaedius: Search for function fields with many rational places

Abstract:

We will give a short introduction to function fields, aimed at providing us with tools to compute L-polynomials of hyperelliptic function fields. We use these tools to conclude the existence of extensions of these function fields for which we can both provide a lower limit on their number of rational places and compute their genus.

Using these techniques we write a program in Java aimed at searching for function fields with a large number of rational places with respect to its genus. Finally we present the results of running the program over various small finite fields and genera.