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Hecke Equivariance of Divisor Lifting with respect to Sesquiharmonic Maass Forms

Daeyeol Jeon, Soon-Yi Kang, Chang Heon Kim

TL;DR

This work extends the divisor lifting framework to sesquiharmonic Maass forms by using the preimages ${\mathbb J}_{1,n}$ of the $J_n$ under the hyperbolic Laplacian. It proves that the divisor lifting is Hecke equivariant up to an explicit constant, giving a $p$-plication formula $\mathbb{J}_{1,n}|T_p = \mathbb{J}_{1,pn}+p\mathbb{J}_{1,n/p}$ (with a corresponding constant term for $n=0$) and showing that ${\mathbb J}_{1,n}$ form a Hecke system. The paper also leverages the Hecke equivariance together with the Borcherds isomorphism to derive twisted-trace identities for singular moduli and to extend Matsusaka-type results to general discriminants, thereby connecting divisor liftings, modular traces, and $L$-values in the sesquiharmonic setting. These results deepen the interaction between polyharmonic Maass forms and arithmetic objects like Dirichlet $L$-functions and class numbers, with potential implications for trace formulas and modular polynomial relations.

Abstract

We investigate the properties of Hecke operator for sesquiharmonic Maass forms. We begin by proving Hecke equivariance of the divisor lifting with respect to sesquiharmonic Mass functions, which maps an integral weight meromorphic modular form to the holomorphic part of the Fourier expansion of a weight 2 sesquiharmonic Maass form. Using this Hecke equivariance, we show that the sesquiharmonic Maass functions, whose images under the hyperbolic Laplace operator are the Faber polynomials $J_n$ of the $j$-function, form a Hecke system analogous to $J_n$. By combining the Hecke equivariance of the divisor lifting with that of the Borcherds isomorphism, we extend Matsusaka's finding on the twisted traces of sesquiharmonic Maass functions.

Hecke Equivariance of Divisor Lifting with respect to Sesquiharmonic Maass Forms

TL;DR

This work extends the divisor lifting framework to sesquiharmonic Maass forms by using the preimages of the under the hyperbolic Laplacian. It proves that the divisor lifting is Hecke equivariant up to an explicit constant, giving a -plication formula (with a corresponding constant term for ) and showing that form a Hecke system. The paper also leverages the Hecke equivariance together with the Borcherds isomorphism to derive twisted-trace identities for singular moduli and to extend Matsusaka-type results to general discriminants, thereby connecting divisor liftings, modular traces, and -values in the sesquiharmonic setting. These results deepen the interaction between polyharmonic Maass forms and arithmetic objects like Dirichlet -functions and class numbers, with potential implications for trace formulas and modular polynomial relations.

Abstract

We investigate the properties of Hecke operator for sesquiharmonic Maass forms. We begin by proving Hecke equivariance of the divisor lifting with respect to sesquiharmonic Mass functions, which maps an integral weight meromorphic modular form to the holomorphic part of the Fourier expansion of a weight 2 sesquiharmonic Maass form. Using this Hecke equivariance, we show that the sesquiharmonic Maass functions, whose images under the hyperbolic Laplace operator are the Faber polynomials of the -function, form a Hecke system analogous to . By combining the Hecke equivariance of the divisor lifting with that of the Borcherds isomorphism, we extend Matsusaka's finding on the twisted traces of sesquiharmonic Maass functions.
Paper Structure (8 sections, 8 theorems, 69 equations)

This paper contains 8 sections, 8 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. preliminaries
  3. Sesquiharmonic Maass Forms
  4. Hecke operators
  5. Hecke equivariance of the Borcherds Isomorphism
  6. Proof of Theorem \ref{['main-HE']}
  7. Proof of Corollaries \ref{['app2']} and \ref{['app3']}
  8. Proof of Corollaries \ref{['cong']} and \ref{['app1']}

Key Result

Theorem 1.1

Let $f$ be a meromorphic modular form for $\Gamma$. Let $p$ be a prime and $\mathcal{T}(p)$ denote the multiplicative Hecke operator acting on the multiplicative group of integer weight meromorphic modular forms for $\Gamma$, as defined in mtp. Let $h := {\rm ord}_{\infty}(f)$ and assume that ${\rm

Theorems & Definitions (10)

  • Theorem 1.1
  • Corollary 1.2: $p$-plication formula
  • Corollary 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • proof : Proof of Corollary \ref{['app2']}
  • proof : Proof of Corollary \ref{['app3']}