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A weighted vertical Sato-Tate law for Maaß forms on $\rm{GSp}_4$

Félicien Comtat

TL;DR

<3-5 sentence high-level summary>This paper establishes a weighted vertical Sato-Tate law for Satake parameters of Maaß forms on GSp4 in the level aspect, proving that a natural spectral-weighted average of Satake parameters converges to the GSp4 Sato-Tate measure as the level grows. The authors reduce the problem to the Kuznetsov formula on GSp4, reduce and bound the continuous spectrum via a detailed analysis of double cosets P(A_f)ackslash G(A_f)/H, and compute ramified local integrals for all ramified parabolic subgroups. They also treat the Archimedean contributions through explicit Jacquet- and Whittaker-type integrals and establish lower bounds for L-values needed to control the spectral weights. The results yield a cuspidal refinement in the squarefree level case and illustrate deep interactions between global automorphic spectral data and local representation theory in higher rank.

Abstract

We prove a weighted Sato-Tate law for the Satake parameters of automorphic forms on $\rm{GSp}_4$ with respect to a fairly general congruence subgroup $H$ whose level tends to infinity. When the level is squarefree we refine our result to the cuspidal spectrum. The ingredients are the $\rm{GSp}_4$ Kuznetsov formula and the explicit calculation of local integrals involved in the Whittaker coefficients of $\rm{GSp}_4$ Eisenstein series. We also discuss how the problem of bounding the continuous spectrum in the level aspect naturally leads to some combinatorial questions involving the double cosets in $P \backslash G / H$, for each parabolic subgroup $P$.

A weighted vertical Sato-Tate law for Maaß forms on $\rm{GSp}_4$

TL;DR

<3-5 sentence high-level summary>This paper establishes a weighted vertical Sato-Tate law for Satake parameters of Maaß forms on GSp4 in the level aspect, proving that a natural spectral-weighted average of Satake parameters converges to the GSp4 Sato-Tate measure as the level grows. The authors reduce the problem to the Kuznetsov formula on GSp4, reduce and bound the continuous spectrum via a detailed analysis of double cosets P(A_f)ackslash G(A_f)/H, and compute ramified local integrals for all ramified parabolic subgroups. They also treat the Archimedean contributions through explicit Jacquet- and Whittaker-type integrals and establish lower bounds for L-values needed to control the spectral weights. The results yield a cuspidal refinement in the squarefree level case and illustrate deep interactions between global automorphic spectral data and local representation theory in higher rank.

Abstract

We prove a weighted Sato-Tate law for the Satake parameters of automorphic forms on with respect to a fairly general congruence subgroup whose level tends to infinity. When the level is squarefree we refine our result to the cuspidal spectrum. The ingredients are the Kuznetsov formula and the explicit calculation of local integrals involved in the Whittaker coefficients of Eisenstein series. We also discuss how the problem of bounding the continuous spectrum in the level aspect naturally leads to some combinatorial questions involving the double cosets in , for each parabolic subgroup .
Paper Structure (50 sections, 76 theorems, 413 equations)

This paper contains 50 sections, 76 theorems, 413 equations.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Parabolic subgroups and their induced representations
  4. Power functions
  5. Eisenstein series and induced representations
  6. Spectral parameters
  7. Whittaker coefficients
  8. Factorization of the Whittaker coefficients
  9. Example: $\mathop{\mathrm{GL}}\nolimits_2$
  10. Whittaker coefficients of Eisenstein series
  11. Miscellaneous lemmas
  12. Group-theoretic lemmas
  13. An integration formula
  14. Support of the Whittaker functions
  15. Equidistribution over the whole spectrum
  16. ...and 35 more sections

Key Result

Theorem 1.1

Let $K(N) \subset \mathop{\mathrm{GSp}}\nolimits_4(\mathbb{A}_\text{fin})$ be a compact open subgroup satisfying Assumption GeneralAssumption. For instance $K(N)$ can be the Borel, Siegel, Klingen or paramodular subgroup of level $N$. Consider the weight $w_{K(N)}$ defined in (weigth). Then for any where $d\mu_{\text{ST}}$ is the Sato-Tate measure for $\mathop{\mathrm{GSp}}\nolimits_4$ and $\int

Theorems & Definitions (158)

  • Theorem 1.1: Theorem \ref{['Thm1']}
  • Theorem 1.2: Theorem \ref{['Thm2']}
  • Remark 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 148 more