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Relative Langlands Duality of Toric Periods

Eric Y. Chen

TL;DR

The paper develops a toric incarnation of relative Langlands duality, focusing on affine toric varieties to illuminate how automorphic X-periods and spectral L-functions can be matched under dual Hamiltonian actions, even in singular settings. It introduces graded toric duality and proves a weak numerical duality for affine toric pairs, with a discrepancy controlled by the rank and the G_gr weight; it further refines this via orbit-by-orbit regularization to connect all orbit contributions and extends the framework to toric Deligne–Mumford stacks. The approach unifies automorphic and spectral perspectives through fixed-point Euler products and a combinatorial duality of cones, offering a coherent picture that also ties into regularization mechanisms and broader questions such as Manin-type height conjectures for toric varieties. The results provide a concrete, combinatorial pathway to understanding nonabelian L-functions in a relative Langlands setting and suggest natural extensions to stacks and metaplectic covers with potential arithmetic applications.

Abstract

The relative Langlands program introduced by Ben-Zvi--Sakellaridis--Venkatesh posits a duality structure exchanging automorphic periods and L-functions, which can be encoded by pairs of dual Hamiltonian actions. In work of the author and Venkatesh, an extension of the definitions to certain singular spaces was made with the objective of restoring duality in some well-known automorphic integrals. In this companion article we apply these definitions to establish duality in the context of affine toric varieties, and study finer structures regarding regularization and stabilizers that are instructive for the general case.

Relative Langlands Duality of Toric Periods

TL;DR

The paper develops a toric incarnation of relative Langlands duality, focusing on affine toric varieties to illuminate how automorphic X-periods and spectral L-functions can be matched under dual Hamiltonian actions, even in singular settings. It introduces graded toric duality and proves a weak numerical duality for affine toric pairs, with a discrepancy controlled by the rank and the G_gr weight; it further refines this via orbit-by-orbit regularization to connect all orbit contributions and extends the framework to toric Deligne–Mumford stacks. The approach unifies automorphic and spectral perspectives through fixed-point Euler products and a combinatorial duality of cones, offering a coherent picture that also ties into regularization mechanisms and broader questions such as Manin-type height conjectures for toric varieties. The results provide a concrete, combinatorial pathway to understanding nonabelian L-functions in a relative Langlands setting and suggest natural extensions to stacks and metaplectic covers with potential arithmetic applications.

Abstract

The relative Langlands program introduced by Ben-Zvi--Sakellaridis--Venkatesh posits a duality structure exchanging automorphic periods and L-functions, which can be encoded by pairs of dual Hamiltonian actions. In work of the author and Venkatesh, an extension of the definitions to certain singular spaces was made with the objective of restoring duality in some well-known automorphic integrals. In this companion article we apply these definitions to establish duality in the context of affine toric varieties, and study finer structures regarding regularization and stabilizers that are instructive for the general case.
Paper Structure (33 sections, 7 theorems, 106 equations)

This paper contains 33 sections, 7 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Relative Langlands duality
  3. Singular spaces
  4. Regularization
  5. Disconnected stabilizers
  6. Notation
  7. Tori
  8. Coefficient fields
  9. Group actions
  10. Curves
  11. Local notation
  12. Galois parameters
  13. Spin structures
  14. Measures
  15. Normalization of class field theory
  16. ...and 18 more sections

Key Result

Theorem 2.1

Let $X$ be the toric variety associated to the fan $\Sigma$ in $X_*(T)_\mathbf{R}$. Then there is a bijective correspondence where $S_\sigma \subset T$ is the subtorus with cocharacter lattice generated by $\sigma \cap X_*(T)$, with the following properties (we write $\tau \leq \sigma$ if $\tau$ is a cone in $\Sigma$ containing $\sigma$ as a face):

Theorems & Definitions (38)

  • Definition 1.1
  • Theorem 2.1: Theorem 3.2.6 CLS
  • Theorem 2.2: Proposition 3.2.7 CLS
  • Definition 2.3
  • Remark 3.1: Convergence of the regularized automorphic period
  • Remark 3.2: Convergence of the spectral period
  • Definition 3.3
  • Theorem 4.1
  • Lemma 4.2
  • proof
  • ...and 28 more