Quadratic forms and their theta series - infinitesimal aspects

TL;DR

This paper investigates the properties of the theta map associated with real quadratic forms, introducing invariants that determine when its differential vanishes or degenerates, with examples and modular form connections.

Contribution

It introduces two invariants for the theta map's differential, demonstrating their modular nature and providing explicit examples in rank two cases.

Findings

Invariants reflect the differential's behavior of the theta map.

Examples of lattices with zero differential are provided.

Invariants are shown to be modular forms for integral lattices.

Abstract

We study the theta map which assigns to a real quadratic form its theta series. We introduce two invariants reflecting whether the differential of the theta map vanishes or is degenerate. We provide examples of lattices where this differential is zero. These invariants turn out to be modular forms for integral lattices. We illustrate this in the rank two case.