Vector valued theta functions associated with binary quadratic forms

TL;DR

This paper investigates vector valued theta functions linked to positive definite even lattices of rank two, establishing their relation to scalar theta functions, constructing an orthogonal basis, and providing explicit Petersson norm formulas.

Contribution

It introduces a detailed analysis of vector valued theta functions for rank two lattices, connecting them to scalar functions and explicitly computing their Petersson norms.

Findings

Established the relation between vector and scalar theta functions.

Constructed an explicit orthogonal basis for the space.

Derived formulas for Petersson norms of basis elements.

Abstract

We study the space of vector valued theta functions for the Weil representation of a positive definite even lattice of rank two with fundamental discriminant. We work out the relation of this space to the corresponding scalar valued theta functions of weight one and determine an orthogonal basis with respect to the Petersson inner product. Moreover, we give an explicit formula for the Petersson norms of the elements of this basis.