Automorphic vector-forms using the Cohn-Elkies magic functions

TL;DR

This paper develops the theory of Hecke vector-forms, which are vector function analogues of automorphic forms, providing a new way to understand quasiautomorphic forms on Hecke triangle groups and relating them to differential equations.

Contribution

It introduces Hecke vector-forms for quasiautomorphic forms on arbitrary Hecke triangle groups, establishing their transformation properties and connections to differential equations.

Findings

Hecke vector-forms transform like automorphic forms on Hecke groups.

The proof relies on properties of binomial coefficients.

Connections to Hecke automorphic linear differential equations.

Abstract

In this study, we introduce the theory of what we call Hecke vector-forms. A Hecke vector-form can be viewed as a vector function representation of some quasiautomorphic form that transforms like an automorphic form on an arbitrarily chosen Hecke triangle group. In other words, because quasiautomorphic forms have complicated transformation behavior when compared with automorphic forms, the construction of a Hecke vector-form is to retrieve a transformation behavior analogous to the simpler, automorphic case. In this way, a Hecke vector-form can be viewed as the vector function analogue of an automorphic form. Since our work is for any quasi-automorphic form over an arbitrary Hecke triangle group, we briefly review the construction of such groups. Furthermore, we review the derivation of the hauptmodul, the automorphic forms, and the normalized quasiautomorphic form of weight 2 for any Hecke triangle group. We then proceed to the theory of Hecke vector-forms and establish the desired transformation behavior with respect to the generators of the associated group. A proof of this fact is strictly elementary, relying on fine properties of binomial coefficients. Lastly, we relate the vector-forms to Hecke automorphic linear differential equations, which are analogues of the frequently researched modular linear differential equations. Our results include Hecke vector-forms of the classical quasimodular forms as the simplest case.