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Higher Period Integrals and Derivatives of L-functions

Shurui Liu, Zeyu Wang

Abstract

We propose a geometric framework to produce a formula relating higher period integrals to higher central derivatives of $L$-functions over function fields, extending the framework of relative Langlands duality à la Ben-Zvi--Sakellaridis--Venkatesh to higher derivatives. For a strongly tempered affine smooth $G$-variety $X$, we give a geometric construction of the action of $L$-observables on the geometric period integral of a Hecke eigensheaf. By taking a suitable version of Frobenius trace of this action, we recover higher central derivatives of the $L$-function attached to the dual symplectic representation. As an application, in the Rankin--Selberg case $(\mathrm{GL}_n\times\mathrm{GL}_{n-1},\mathrm{GL}_{n-1})$, we obtain a formula for higher derivatives of the Rankin--Selberg $L$-function. This provides a conceptual generalization of Yun--Zhang's higher Gross--Zagier formula to higher-dimensional spherical varieties.

Higher Period Integrals and Derivatives of L-functions

Abstract

We propose a geometric framework to produce a formula relating higher period integrals to higher central derivatives of -functions over function fields, extending the framework of relative Langlands duality à la Ben-Zvi--Sakellaridis--Venkatesh to higher derivatives. For a strongly tempered affine smooth -variety , we give a geometric construction of the action of -observables on the geometric period integral of a Hecke eigensheaf. By taking a suitable version of Frobenius trace of this action, we recover higher central derivatives of the -function attached to the dual symplectic representation. As an application, in the Rankin--Selberg case , we obtain a formula for higher derivatives of the Rankin--Selberg -function. This provides a conceptual generalization of Yun--Zhang's higher Gross--Zagier formula to higher-dimensional spherical varieties.

Paper Structure

This paper contains 110 sections, 50 theorems, 455 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Period integrals and special values of $L$-functions
  4. Special cycles and higher derivatives of $L$-functions
  5. A conceptual and uniform framework in the strongly tempered case
  6. Main result: a general formulation
  7. Geometric result
  8. Arithmetic result
  9. Example: Rankin--Selberg case
  10. Geometric result
  11. Arithmetic result
  12. Outline of the proof
  13. Linear algebra: Kolyvagin systems
  14. Commutator relations on cohomological correspondences: local-to-global
  15. Automorphic Clifford relations
  16. ...and 95 more sections

Key Result

Theorem 1.2

\newlabelclifgen Assuming BZSV, there exists an action of $\operatorname{Cl}(M)$ on $\int_X \mathbb{L}_{\sigma}$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (174)

  • Example 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6: \ref{['higherrsbzsvglobal']}
  • Theorem 1.7: \ref{['higherformulagln']}
  • Remark 1.8
  • Definition 3.2
  • Definition 3.3
  • ...and 164 more