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Integral structures in smooth $\mathrm{GL}_2(\mathbf{Q}_p)$-representations and zeta integrals

Alexandros Groutides

TL;DR

The paper develops an integral, lattice-theoretic framework for spherical GL$_2$(\mathbf{Q}_p) representations and their interaction with toric periods under an unramified torus H. By placing these representations in a universal Hecke-module setting and employing zeta-integrals and Shintani techniques, it proves an integral analogue of multiplicity one and establishes optimal, explicitly controlled norm-relations linked to local $L$-factors; it also extends to toric periods of modular forms, giving integrality statements that interpolate across unramified data via universal objects. The results yield a robust arithmetic refinement of Waldspurger-type toric period phenomena and provide explicit formulas tying Shintani data to Satake parameters and local Euler factors, with split and non-split torus cases treated distinctly. Overall, the work advances the integral theory in the relative Langlands program for GL$_2$ and supplies tools for arithmetic applications to Euler systems and modular forms.

Abstract

Using zeta-integrals and lattices of functions on a spherical variety, we study integral structures in spherical representations of $\mathrm{GL}_2(\mathbf{Q}_p)$ and their interaction with the unique linear functional invariant under an unramified maximal torus. Within this framework, we reformulate and prove the first instance of optimality of abstract integral norm-relations as proposed by Loeffler. We also interpret this as a form of integrality for toric periods associated to modular forms, where part of it can be regarded as an arithmetic integral analogue of Waldspurger's multiplicity one in the unramified setting.

Integral structures in smooth $\mathrm{GL}_2(\mathbf{Q}_p)$-representations and zeta integrals

TL;DR

The paper develops an integral, lattice-theoretic framework for spherical GL(\mathbf{Q}_p) representations and their interaction with toric periods under an unramified torus H. By placing these representations in a universal Hecke-module setting and employing zeta-integrals and Shintani techniques, it proves an integral analogue of multiplicity one and establishes optimal, explicitly controlled norm-relations linked to local -factors; it also extends to toric periods of modular forms, giving integrality statements that interpolate across unramified data via universal objects. The results yield a robust arithmetic refinement of Waldspurger-type toric period phenomena and provide explicit formulas tying Shintani data to Satake parameters and local Euler factors, with split and non-split torus cases treated distinctly. Overall, the work advances the integral theory in the relative Langlands program for GL and supplies tools for arithmetic applications to Euler systems and modular forms.

Abstract

Using zeta-integrals and lattices of functions on a spherical variety, we study integral structures in spherical representations of and their interaction with the unique linear functional invariant under an unramified maximal torus. Within this framework, we reformulate and prove the first instance of optimality of abstract integral norm-relations as proposed by Loeffler. We also interpret this as a form of integrality for toric periods associated to modular forms, where part of it can be regarded as an arithmetic integral analogue of Waldspurger's multiplicity one in the unramified setting.
Paper Structure (27 sections, 35 theorems, 120 equations)

This paper contains 27 sections, 35 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Acknowledgements
  4. Preliminaries
  5. Group schemes, Hecke algebras and Hecke modules
  6. Hecke modules
  7. From invariants to coinvariants
  8. Double coset decompositions
  9. Universal Hecke modules
  10. Structure theorems
  11. Multiplicity one
  12. Universal Hecke operators
  13. Modules for a fixed character
  14. Universal property
  15. Specialisation at an unramified character
  16. ...and 12 more sections

Key Result

Theorem A

Let $\xi_0$ be the characteristic function of $\mathfrak{H}(\mathbf{Q}_p)\mathfrak{G}(\mathbf{Z}_p)$. The following are true:

Theorems & Definitions (87)

  • Theorem A: \ref{['thm L1 7']}
  • Theorem B: Integrality of toric periods for modular forms; \ref{['thm integr of toric periods']}
  • Definition 2.1.1
  • Proposition 2.1.2
  • proof
  • Lemma 2.1.3
  • proof
  • Lemma 2.1.4
  • proof
  • Definition 2.1.5
  • ...and 77 more