Abstract

The aim of this thesis is to give an efficient method to compute a certain theta function Θ(a,b;z), up to a given p-adic precision n. The function Θ(a,b;z) arises from the Hurwitz quaternions and is meromorphic on the upper-half plane. We will first discuss a ”na¨ıve” method to compute Θ(a,b;z) and, by counting Hurwitz quaternions of a given norm, we will show that this method is not efficient. We will then develop some recursive relations for the Hurwitz quaterions, which will be the fundamental tool to describe a more efficient way to compute Θ(a,b;z).