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Traces of partition Eisenstein series

  • Tewodros Amdeberhan , Michael Griffin , Ken Ono EMAIL logo and Ajit Singh
Published/Copyright: February 11, 2025
Forum Mathematicum
From the journal Forum Mathematicum Volume 37 Issue 6

Abstract

We study “partition Eisenstein series”, extensions of the Eisenstein series G 2 ⁢ k ⁢ ( τ ) , defined by

λ = ( 1 m 1 , 2 m 2 , … , k m k ) ⊢ k ↦ G λ ⁢ ( τ ) := G 2 ⁢ ( τ ) m 1 ⁢ G 4 ⁢ ( τ ) m 2 ⁢ ⋯ ⁢ G 2 ⁢ k ⁢ ( τ ) m k .

For functions ϕ : 𝒫 ↦ ℂ on partitions, the weight 2 ⁢ k partition Eisenstein trace is the quasimodular form

Tr k ⁡ ( ϕ ; τ ) := ∑ λ ⊢ k ϕ ⁢ ( λ ) ⁢ G λ ⁢ ( τ ) .

These traces give explicit formulas for some well-known generating functions, such as the kth elementary symmetric functions of the inverse points of 2-dimensional complex lattices ℤ ⊕ ℤ ⁢ τ , as well as the 2 ⁢ k 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.

Keywords: Partition Eisenstein series; quasimodular forms; Andrews–Garvan crank; Jacobi forms
MSC 2020: 11F03; 05A17; 11M36

Communicated by Jan Bruinier

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-2002265

Award Identifier / Grant number: DMS-2055118

Funding statement: The authors thank the referee, Kathrin Bringmann and Badri Pandey for comments that improved this paper. The third author thanks the Thomas Jefferson Fund and the NSF (DMS-2002265 and DMS-2055118). The fourth author is grateful for the support of a Fulbright Nehru Postdoctoral Fellowship.

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Received: 2024-08-16
Revised: 2025-01-02
Published Online: 2025-02-11
Published in Print: 2025-06-01

© 2025 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Weil representations of twisted loop groups of type A n (2)
  3. Local Birkhoff decompositions for loop groups and a finiteness result
  4. Uniform boundedness of oscillatory singular integrals with rational phases
  5. On the stabilizer of the graph of linear functions over finite fields
  6. Global solution and blow-up of critical heat equation with nonlocal interaction
  7. Locally compact groups with all dense subgroups separable
  8. On the Kawaguchi–Silverman conjecture for birational automorphisms of irregular varieties
  9. On isometry groups of gradient Ricci solitons
  10. Extensions of a theorem of P. Hall on indexes of maximal subgroups
  11. A trace formula for Hecke operators on Fuchsian groups
  12. Quasi-hereditary algebras with all standard modules being isomorphic to submodules of projective modules
  13. Computation of endo-fixed closures in free-abelian times free groups
  14. Non-vanishing of Maass form 𝐿-functions of cubic level at the central point
  15. Traces of partition Eisenstein series
  16. Gabor System based on the unitary dual of the Heisenberg group
  17. Zeros of linear combinations of Dirichlet 𝐿-functions on the critical line
  18. Paley inequality for the Weyl transform and its applications
https://doi.org/10.1515/forum-2024-0388
Keywords for this article
Partition Eisenstein series; quasimodular forms; Andrews–Garvan crank; Jacobi forms

Articles in the same Issue

  1. Frontmatter
  2. Weil representations of twisted loop groups of type A n (2)
  3. Local Birkhoff decompositions for loop groups and a finiteness result
  4. Uniform boundedness of oscillatory singular integrals with rational phases
  5. On the stabilizer of the graph of linear functions over finite fields
  6. Global solution and blow-up of critical heat equation with nonlocal interaction
  7. Locally compact groups with all dense subgroups separable
  8. On the Kawaguchi–Silverman conjecture for birational automorphisms of irregular varieties
  9. On isometry groups of gradient Ricci solitons
  10. Extensions of a theorem of P. Hall on indexes of maximal subgroups
  11. A trace formula for Hecke operators on Fuchsian groups
  12. Quasi-hereditary algebras with all standard modules being isomorphic to submodules of projective modules
  13. Computation of endo-fixed closures in free-abelian times free groups
  14. Non-vanishing of Maass form 𝐿-functions of cubic level at the central point
  15. Traces of partition Eisenstein series
  16. Gabor System based on the unitary dual of the Heisenberg group
  17. Zeros of linear combinations of Dirichlet 𝐿-functions on the critical line
  18. Paley inequality for the Weyl transform and its applications
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