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Traces of partition Eisenstein series

Tewodros Amdeberhan, Michael Griffin, Ken Ono, Ajit Singh

TL;DR

The paper introduces partition Eisenstein series $G_\lambda(\tau)$ and their traces $\mathrm{Tr}_k(\phi;\tau)$, showing they capture holomorphic parts of symmetric functions of inverse lattice points and the even crank moments via explicit generating functions. By applying Pólya’s cycle index framework to the Weierstrass $\sigma$-function, it proves that the quantities $e_k(\Lambda_\tau(0))$ are expressed as polynomials in Eisenstein series and that their holomorphic parts coincide with partition traces, delivering Theorem 1. It then establishes a deep link between crank moments and these traces (Theorem 2), providing explicit expansions and polynomial representations, including Corollary 1 with concrete examples and Lambert-series connections. Finally, the authors develop a general Jacobi-form framework with torsional divisors (Theorem 3), showing that Taylor coefficients of meromorphic Jacobi forms factor through traces of partition Eisenstein series, thus unifying partition combinatorics, modular objects, and Jacobi form theory in a broad spectral setting.$

Abstract

We study "partition Eisenstein series", extensions of the Eisenstein series $G_{2k}(τ),$ defined by $$λ=(1^{m_1}, 2^{m_2},\dots, k^{m_k}) \vdash k \ \ \ \ \ \longmapsto \ \ \ \ \ G_λ(τ):= G_2(τ)^{m_1} G_4(τ)^{m_2}\cdots G_{2k}(τ)^{m_k}. $$ For functions $φ: \mathcal{P}\rightarrow \mathbb{C}$ on partitions, the weight $2k$ "partition Eisenstein trace" is the quasimodular form $$ {\mathrm{Tr}}_k(φ;τ):=\sum_{λ\vdash k} φ(λ)G_λ(τ). $$ These traces give explicit formulas for some well-known generating functions, such as the $k$th elementary symmetric functions of the inverse points of 2-dimensional complex lattices $\mathbb{Z}\oplus \mathbb{Z}τ,$ as well as the $2k$th power moments of the Andrews-Garvan crank function. To underscore the ubiquity of such traces, we show that their generalizations give the Taylor coefficients of generic Jacobi forms with torsional divisor.

Traces of partition Eisenstein series

TL;DR

The paper introduces partition Eisenstein series and their traces , showing they capture holomorphic parts of symmetric functions of inverse lattice points and the even crank moments via explicit generating functions. By applying Pólya’s cycle index framework to the Weierstrass -function, it proves that the quantities are expressed as polynomials in Eisenstein series and that their holomorphic parts coincide with partition traces, delivering Theorem 1. It then establishes a deep link between crank moments and these traces (Theorem 2), providing explicit expansions and polynomial representations, including Corollary 1 with concrete examples and Lambert-series connections. Finally, the authors develop a general Jacobi-form framework with torsional divisors (Theorem 3), showing that Taylor coefficients of meromorphic Jacobi forms factor through traces of partition Eisenstein series, thus unifying partition combinatorics, modular objects, and Jacobi form theory in a broad spectral setting.$

Abstract

We study "partition Eisenstein series", extensions of the Eisenstein series defined by For functions on partitions, the weight "partition Eisenstein trace" is the quasimodular form These traces give explicit formulas for some well-known generating functions, such as the th elementary symmetric functions of the inverse points of 2-dimensional complex lattices as well as the th power moments of the Andrews-Garvan crank function. To underscore the ubiquity of such traces, we show that their generalizations give the Taylor coefficients of generic Jacobi forms with torsional divisor.
Paper Structure (12 sections, 10 theorems, 88 equations)

This paper contains 12 sections, 10 theorems, 88 equations.

Table of Contents

  1. Introduction and Statement of Results
  2. Pólya's cycle index polynomials
  3. Proof of Theorem \ref{['Theorem1']}
  4. Proof of Theorem \ref{['Theorem2']}
  5. Two lemmas
  6. Proof of Theorem \ref{['Theorem2']}
  7. Proof of Corollary \ref{['Cor1']}
  8. Proof of Theorem \ref{['Theorem3']}
  9. Background on Jacobi forms
  10. A key Jacobi form factorization
  11. Eisenstein series for torsional divisors
  12. Proof of Theorem \ref{['Theorem3']}

Key Result

Theorem 1.1

If $k$ is a positive integer, then the following are true. (1) We have that is a weight $2k$ nonholomorphic modular form on $\mathrm{SL}_2(\mathbb Z).$ (2) There is a degree $k$ isobaric polynomialA polynomial is isobaric of degree $k$ if each of its monomials $X_1^{d_1}X_2^{d_2}\cdots X_k^{d_k}$ satisfies $d_1+2d_2+\dots+kd_k=k.$$F_k(X_1, X_2,\dots, X_k)$ for which where $G_{2}^{\star}(\tau)$ i

Theorems & Definitions (19)

  • Remark
  • Theorem 1.1
  • Remark
  • Theorem 1.2
  • Corollary 1.3
  • Remark
  • Definition
  • Theorem 1.4
  • Lemma 2.1: Example 5.2.10 of Stanley
  • Example
  • ...and 9 more