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Integrality of Hecke eigenvalues and the growth of Hecke fields

Kenji Sakugawa, Shingo Sugiyama

Abstract

We prove that Hecke eigenvalues for any Hilbert and Siegel modular forms are algebraic integers. Our method does not rely on cohomologicality nor Galois representations. We apply the integrality of Hecke eigenvalues for Hilbert modular forms of non-parallel weight to the estimation of the growth of Hecke fields of Hilbert cusp forms with non-vanishing central $L$-values. As a further application, we give the growth of the fields of rationality of cuspidal automorphic representations of ${\rm GL}_{2d}(\mathbb{A}_\mathbb{Q})$ for a prime number $d$ with non-vanishing central $L$-values. We also apply the integrality of Hecke eigenvalues for holomorphic Siegel cusp forms of general degree in order to give the growth of the Hecke fields of those forms.

Integrality of Hecke eigenvalues and the growth of Hecke fields

Abstract

We prove that Hecke eigenvalues for any Hilbert and Siegel modular forms are algebraic integers. Our method does not rely on cohomologicality nor Galois representations. We apply the integrality of Hecke eigenvalues for Hilbert modular forms of non-parallel weight to the estimation of the growth of Hecke fields of Hilbert cusp forms with non-vanishing central -values. As a further application, we give the growth of the fields of rationality of cuspidal automorphic representations of for a prime number with non-vanishing central -values. We also apply the integrality of Hecke eigenvalues for holomorphic Siegel cusp forms of general degree in order to give the growth of the Hecke fields of those forms.
Paper Structure (23 sections, 47 theorems, 163 equations)

This paper contains 23 sections, 47 theorems, 163 equations.

Table of Contents

  1. Introduction
  2. Case of Hilbert modular forms
  3. Case of Siegel modular forms
  4. Applications to the growth of the fields of rationality and the non-vanishing of central $L$-values
  5. Growth of Hecke fields of Siegel modular forms of degree 2
  6. Growth of Hecke fields of Siegel modular forms of general degree
  7. Organization of this paper
  8. Notation
  9. Integrality of eigenvalues
  10. Abstract setting
  11. Proof of Theorem \ref{['thm0.1']}
  12. Proof of Theorem \ref{['integrality for SMF']}
  13. Applications of integrality to the growth of Hecke fields
  14. Hilbert modular case
  15. Proof of Corollaries \ref{['Hecke fields for degree 2']} and \ref{['Hecke fields for degree n']} (Siegel modular case)
  16. ...and 8 more sections

Key Result

Theorem 1.1

All eigenvalues of any Hecke operator $T\in {\bf T}_{F, \mathbb{Z}}^{{\bf k},{\mathfrak{n}},\chi}$ on $M_{\bf k}(\Gamma_0({\mathfrak{n}}), \chi)$ are algebraic integers.

Theorems & Definitions (96)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3: Precisely, see Corollary \ref{['nonzero L and growth of Hecke']}
  • Corollary 1.4: Precisely, see Corollary \ref{['lower bound for Hecke field']}
  • Corollary 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • ...and 86 more