On multiplier systems and theta functions of half-integral weight for the Hilbert modular group $\mathrm{SL}_2(\mathfrak{o})$

TL;DR

This paper investigates the existence and properties of theta functions of half-integral weight as Hilbert modular forms over totally real fields, establishing conditions for their existence and associated multiplier systems.

Contribution

It provides an equivalent condition for the existence of multiplier systems of half-integral weight and characterizes when such theta functions exist for the Hilbert modular group.

Findings

Established an equivalent condition for multiplier systems of half-integral weight.

Determined the conditions on the field $F$ for the existence of such theta functions.

Defined theta functions via sums over fractional ideals of $F$.

Abstract

Let $F$ be a totally real number field and $\mathfrak{o}$ the ring of integers of $F$. We study theta functions which are Hilbert modular forms of half-integral weight for the Hilbert modular group $\mathrm{SL}_2(\mathfrak{o})$. We obtain an equivalent condition that there exists a multiplier system of half-integral weight for $\mathrm{SL}_2(\mathfrak{o})$. We determine the condition of $F$ that there exists a theta function which is a Hilbert modular form of half-integral weight for $\mathrm{SL}_2(\mathfrak{o})$. The theta function is defined by a sum on a fractional ideal $\mathfrak{a}$ of $F$.