Hecke equivariance of the divisor map

TL;DR

This paper demonstrates that the divisor map on modular forms is compatible with Hecke operators, revealing new insights into their algebraic structure and implications for divisor sum formulas and Maass forms.

Contribution

It establishes the Hecke equivariance of the divisor map on modular forms, connecting Hecke actions with divisor theory and extending to polyharmonic Maass forms.

Findings

Divisor map is Hecke equivariant.

Hecke operators relate to divisor sum formulas.

Implications for self-adjointness of Hecke operators.

Abstract

We study the multiplicative Hecke operators acting on the space of meromorphic modular forms, and show that the divisor map to divisors on $X_0(N)$ is a Hecke equivariant map. As applications, we investigate the divisor sum formula of Bruinier-Kohnen-Ono and more general Rohrlich-type divisor sums for polyharmonic Maass forms, discussing several implications for the Hecke action and its relation to the self-adjointness of the Hecke operators.