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Central limit theorem for Hecke eigenvalues

Henry H. Kim, Satoshi Wakatsuki, Takuya Yamauchi

TL;DR

The paper proves a universal central limit theorem for Hecke eigenvalues across a broad class of split simple groups by leveraging irreducible characters of compact Lie groups and the Satake/isotypic framework. It introduces and uses simultaneous vertical Sato-Tate distributions to relate local Satake parameters to global eigenvalue sums over families of cusp forms (Maass, holomorphic, quaternionic, Siegel) and Langlands ${L}$-functions. It provides explicit asymptotics and leading terms in weight, level, and Siegel contexts, and extends the CLT to the exceptional group ${G_2}$ with a degree-seven standard ${L}$-function. The results unify and extend previous CLTs (e.g., for ${PGL_n}$) by a common orthogonality structure of irreducible characters and a robust adelic/Sato-Tate analysis, with potential applications to fluctuations of automorphic data in large families.

Abstract

In this paper, we obtain the central limit theorem of Hecke eigenvalues in very general setting of split simple algebraic groups over $\mathbb{Q}$, using irreducible characters of compact Lie groups.

Central limit theorem for Hecke eigenvalues

TL;DR

The paper proves a universal central limit theorem for Hecke eigenvalues across a broad class of split simple groups by leveraging irreducible characters of compact Lie groups and the Satake/isotypic framework. It introduces and uses simultaneous vertical Sato-Tate distributions to relate local Satake parameters to global eigenvalue sums over families of cusp forms (Maass, holomorphic, quaternionic, Siegel) and Langlands -functions. It provides explicit asymptotics and leading terms in weight, level, and Siegel contexts, and extends the CLT to the exceptional group with a degree-seven standard -function. The results unify and extend previous CLTs (e.g., for ) by a common orthogonality structure of irreducible characters and a robust adelic/Sato-Tate analysis, with potential applications to fluctuations of automorphic data in large families.

Abstract

In this paper, we obtain the central limit theorem of Hecke eigenvalues in very general setting of split simple algebraic groups over , using irreducible characters of compact Lie groups.
Paper Structure (22 sections, 20 theorems, 106 equations)

This paper contains 22 sections, 20 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Unramified Hecke algebra and Sato-Tate measure
  3. Notation
  4. Unramified Hecke algebra
  5. Plancherel measure
  6. Sato-Tate measure
  7. Irreducible characters of compact Lie groups
  8. Satake transform
  9. Asymptotic behaviors of Hecke eigenvalues; Simultaneous vertical Sato-Tate distribution
  10. Notations
  11. Space of cusp forms
  12. A Family of Maass cusp forms
  13. A Family of holomorphic and quaternionic cusp forms; weight aspect
  14. Level aspect for Siegel cusp forms
  15. Central limit theorem
  16. ...and 7 more sections

Key Result

Lemma 2.1

There exists constants $a$, $b\in{\mathbb{N}}$ such that, for any prime $p$ and any $h\in \mathcal{H}^\mathrm{ur}_\kappa(G_p)$, we have

Theorems & Definitions (41)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Corollary 2.4
  • proof
  • Corollary 2.5
  • proof
  • Remark 2.6
  • ...and 31 more