Heat equation for theta functions and vector-valued modular forms

TL;DR

This paper introduces a new method to construct vector-valued modular forms from scalar-valued ones and proves the equivalence of two seemingly different spaces of such forms, linking them through the heat equation for theta functions.

Contribution

A novel construction method for vector-valued modular forms and a proof that two distinct spaces of these forms are actually identical.

Findings

Proved the equality of $V_{grad}$ and $V_ heta$ spaces of modular forms.

Established a connection between the new construction and the heat equation for theta functions.

Demonstrated the equivalence of forms constructed via gradients of theta functions and second order theta constants.

Abstract

We give a new method for constructing vector-valued modular forms from singular scalar-valued ones. As an application we prove the identity between two remarkable spaces of vector-valued modular forms which seem to be unrelated at a first look, since they are constructed in two very different ways. If $V_{grad}$ is the vector space generated by vector-valued modular forms constructed with gradients of odd theta functions and $V_\Theta$ is the one generated by vector-valued modular forms arising from second order theta constants with our new construction, we will prove that $V_{grad}=V_\Theta$. This result could also be proven as a consequence of the "heat equation" for theta functions.