Hilbert modular forms of weight 1/2 and theta functions

TL;DR

This paper extends the classical results of Serre and Stark on modular forms of weight 1/2 to the setting of Hilbert modular forms over totally real fields, addressing new technical challenges.

Contribution

It generalizes the basis construction for weight 1/2 modular forms to Hilbert modular forms over totally real fields with narrow class number 1.

Findings

Established a basis for Hilbert modular forms of weight 1/2 using theta functions.

Overcame technical difficulties in extending classical methods beyond Q.

Provided conditions on level and character for the generalization.

Abstract

Serre and Stark found a basis for the space of modular forms of weight 1/2 in terms of theta series. In this paper, we generalize their result - under certain mild restrictions on the level and character - to the case of weight 1/2 Hilbert modular forms over a totally real field of narrow class number 1. The methods broadly follow those of Serre-Stark; however we are forced to overcome technical difficulties which arise when we move out of Q.