Hecke equivariance of the divisor map

TL;DR

The paper develops a unified Hecke-theoretic framework for meromorphic modular forms by treating three representations of the Hecke algebra: on divisors of X_0(N), on modular forms, and on the multiplicative group of meromorphic modular forms. It proves the divisor map is Hecke-equivariant, i.e., $T(n)\mathrm{div}(f)=\mathrm{div}(f|_*T(n))$ for $(n,N)=1$, linking divisor data to Hecke actions and enabling a reinterpretation of Bruinier–Kohnen–Ono-type pairings via divisor theory. Building on this, the authors introduce Rohrlich-type divisor sums using Niebur–Poincaré series and derive $p$-multiplication (p-lication) formulas for polyharmonic Maass forms, which in turn yield algebraic and analytic consequences, including new proofs and simplifications of results like those of Duke–Imamoğlu–Tóth and related identities. The results illuminate the interplay between Hecke algebras, divisor arithmetic on modular curves, and special value formulas, with potential applications to self-adjointness properties and broader divisor-sum identities. The framework thus provides a robust, algebraic route to previously analytic divisor-sum identities and their Hecke symmetries.

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.

Theorems & Definitions (50)

• Theorem 1.1: \ref{['thm:Hecke-equivariance']}
• Theorem 1.2: \ref{['prop:divisor-sums']}
• Corollary 1.3
• proof
• Definition 2.1
• Example 1
• Definition 2.2
• Proposition 2.3: Shimura Shimura1971
• Lemma 2.4: DiamondShurman2005
• Definition 2.5