Siegel theta series for quadratic forms of signature $(m-1,1)$

TL;DR

This paper explores Siegel theta series for quadratic forms with signature (m-1,1), constructing both holomorphic and non-holomorphic series with different modular properties, and analyzing their relationship.

Contribution

It introduces new constructions of holomorphic and non-holomorphic Siegel theta series for signature (m-1,1) quadratic forms and studies their transformation behaviors.

Findings

Holomorphic series does not transform as a modular form.

Non-holomorphic series transforms as a Siegel modular form.

Holomorphic series approximates the holomorphic part of the modular series almost everywhere.

Abstract

We investigate Siegel theta series for quadratic forms of signature $(m-1,1)$. On the one hand, we construct a holomorphic series that does not transform like a modular form. On the other hand, we construct a non-holomorphic series that transforms like a Siegel modular form of weight $m/2$. Moreover, the holomorphic series describes almost everywhere the holomorphic part of the modular series.