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On triple product $L$-functions and the fiber bundle method

Jayce R. Getz, Miao Pam Gu, Chun-Hsien Hsu, Spencer Leslie

TL;DR

This work builds multivariable zeta integrals whose Euler products encode the triple product $L$-function together with auxiliary $L$-factors, situating the construction in a geometric framework of affine $oldsymbol{}$-bundles. A generalized Poisson summation conjecture is formulated for these spaces, and a fiber bundle strategy is proposed to reduce the global problem to tractable local and linear cases; a follow-up work GGHL2 is announced to complete the local analysis. The paper develops a robust function-theoretic and representation-theoretic apparatus—including Schwartz spaces, eigenmeasures, and line-bundle twists—along with local zeta integrals, including unramified computations that recover the expected $L$-factor structure. If the global Poisson summation conjecture holds (and the local hypotheses are resolved), this framework yields meromorphic continuation and functional equations for triple product $L$-functions, at least for $r_i>2$, marking a significant advance toward Langlands functoriality in higher rank. The fiber-bundle method provides a concrete roadmap for reducing global questions to local verifications on simpler geometric pieces, suggesting new avenues to attack long-standing conjectures via spectral and geometric techniques.

Abstract

We introduce multi-variable zeta integrals which unfold to Euler products representing the triple product $L$-function times a product of $L$-functions with known analytic properties. We then formulate a generalization of the Poisson summation conjecture and show how it implies the analytic properties of triple product $L$-functions. Finally, we propose a strategy, the fiber bundle method, to reduce this generalized conjecture to a simpler case of the Poisson summation conjecture along with certain local compatibility statements.

On triple product $L$-functions and the fiber bundle method

TL;DR

This work builds multivariable zeta integrals whose Euler products encode the triple product -function together with auxiliary -factors, situating the construction in a geometric framework of affine -bundles. A generalized Poisson summation conjecture is formulated for these spaces, and a fiber bundle strategy is proposed to reduce the global problem to tractable local and linear cases; a follow-up work GGHL2 is announced to complete the local analysis. The paper develops a robust function-theoretic and representation-theoretic apparatus—including Schwartz spaces, eigenmeasures, and line-bundle twists—along with local zeta integrals, including unramified computations that recover the expected -factor structure. If the global Poisson summation conjecture holds (and the local hypotheses are resolved), this framework yields meromorphic continuation and functional equations for triple product -functions, at least for , marking a significant advance toward Langlands functoriality in higher rank. The fiber-bundle method provides a concrete roadmap for reducing global questions to local verifications on simpler geometric pieces, suggesting new avenues to attack long-standing conjectures via spectral and geometric techniques.

Abstract

We introduce multi-variable zeta integrals which unfold to Euler products representing the triple product -function times a product of -functions with known analytic properties. We then formulate a generalization of the Poisson summation conjecture and show how it implies the analytic properties of triple product -functions. Finally, we propose a strategy, the fiber bundle method, to reduce this generalized conjecture to a simpler case of the Poisson summation conjecture along with certain local compatibility statements.
Paper Structure (50 sections, 61 theorems, 420 equations)

This paper contains 50 sections, 61 theorems, 420 equations.

Table of Contents

  1. Introduction
  2. The Poisson summation conjecture
  3. Affine $\Psi$-bundles
  4. The integral representation
  5. Local integrals
  6. The fiber bundle method
  7. Outline
  8. Function-theoretic preliminaries
  9. Quasi-characters and norms
  10. Measures
  11. Function spaces
  12. Global setting
  13. Affine $\Psi$-bundles
  14. Systems of basic sections
  15. The case of vector spaces
  16. ...and 35 more sections

Key Result

Theorem 1.1

Assume $f \in C^\infty(X^\circ(\mathbb{A}_F) \times Y^{\circ}(\mathbb{A}_F),\mathcal{L}_{\psi})$ satisfies the assumptions at the beginning of § sec:conv. For $(\mathrm{Re}(\underline{s}),\mathrm{Re}(s))$ in a suitable nonempty cone in $\mathbb{R}^{\underline{1}}_{>0} \times \mathbb{R}_{>0}$ the int where the inner integral is over $U_{\underline{r-2}}(\mathbb{A}_F) \backslash \mathrm{GL}_{\underl

Theorems & Definitions (117)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Lemma 2.1
  • Remark
  • Lemma 2.2
  • proof
  • Remark
  • Lemma 2.3
  • ...and 107 more