Symmetries for Siegel Theta Functions, Borcherds Lifts and Automorphic Green Functions

TL;DR

This paper investigates symmetries of Siegel theta functions associated with quadratic forms and applies these symmetries to automorphic forms, Borcherds lifts, and Green functions related to orthogonal groups.

Contribution

It establishes new symmetry properties of Siegel theta functions and applies them to automorphic forms, Borcherds lifts, and Green functions in the context of quadratic forms.

Findings

Proves symmetry properties of Siegel theta functions.

Demonstrates symmetry relations for automorphic forms on orthogonal groups.

Connects these symmetries to Borcherds lifts and automorphic Green functions.

Abstract

Let q be an integral quadratic form of signature (2,m+2). We will show that the Siegel theta functions attached to q satisfies certain symmetries. As an application, we prove the symmetries for automorphic forms on the orthogonal group of q closely related to Heegener divisors (Borcherds lifts and automorphic Green functions).