Span of Restriction of Hilbert Theta Functions

TL;DR

This paper investigates the restrictions of Hilbert theta series for totally real fields, demonstrating their spanning properties for quadratic and cubic fields, and providing evidence for higher-degree fields.

Contribution

It proves that diagonal restrictions of Hilbert theta series span the space of elliptic modular forms for quadratic and cubic fields, with supporting evidence for higher degrees.

Findings

Restrictions span the modular forms space for quadratic and cubic fields.

Evidence suggests similar spanning phenomena for quartic, quintic, and sextic fields.

The study advances understanding of Hilbert theta series and their relation to modular forms.

Abstract

In this paper, we study the diagonal restrictions of certain Hilbert theta series for a totally real field $F$, and prove that they span the corresponding space of elliptic modular forms when the $F$ is quadratic or cubic. Furthermore, we give evidence of this phenomenon when $F$ is quartic, quintic and sextic.