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Hecke Operators and Systems of Eigenvalues on Siegel Cusp Forms

About this Title

Kazuyuki Hatada

Publication: Memoirs of the American Mathematical Society
Publication Year: 2020; Volume 268, Number 1306
ISBNs: 978-1-4704-4334-4 (print); 978-1-4704-6343-4 (online)
DOI: https://doi.org/10.1090/memo/1306
Published electronically: May 21, 2021
Keywords: Siegel cusp forms, action of Hecke rings on Siegel modular varieties, rigid analytic varieties, estimation of eigenvalues of Hecke operators, the generalized Ramanujan conjecture, $p$-parameters of Siegel cusp eigenforms

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Table of Contents

Chapters

  • 1. Introduction
  • 2. Action of Double Cosets
  • 3. Satake’s Isomorphisms and Hecke Operators
  • 4. Estimates for All the Eigenvalues of Hecke Operators on Siegel Cusp Forms
  • 5. The Generalized Ramanujan Conjecture in Siegel Modular Case
  • 6. $p$-Parameters of Siegel Cusp Eigenforms
  • 7. Some Applications
  • 8. Langlands’ L-Parameters
  • Appendix
  • Index of Theorems, Propositions, Diagrams, Lemmas and Corollaries
  • Index of Notations

Abstract

Let $g$ and ${\mathcal {N}}$ be arbitrary positive integers and let $p$ be any prime number with $p \nmid {\mathcal {N}}.$ Let $\Gamma _{g}({\mathcal {N}})$ denote the principal congruence subgroup of level ${\mathcal {N}}$ of $Sp(g,\mathbb {Z})$($\subset GL(2g,\mathbb {Z})$). Let $\mathcal {S}_{m}(\Gamma _{g}({\mathcal {N}}))$ denote the space of holomorphic Siegel cusp forms of any weight $m \ge g+1$ on $\Gamma _{g}({\mathcal {N}})$. Here we write our main results roughly. We analyse the action of Hecke rings on Siegel modular varieties of arbitrary degrees and arbitrary levels$\ge 3$ using arithmetic toroidal compactifications and rigid analytic spaces, $\ell$-adic cohomology and $p$-adic Hodge theory. We give new congruence relations for Hecke correspondences on Siegel modular varieties. We express Hecke operators $T_m(p)$ and $T_m(p^2,j)$ with $0 \le j \le g$ acting on ${\mathcal {S}}_m(\Gamma _g({\mathcal {N}}))$ by endomorphisms of certain rigid analytic varieties. We give estimation of any Archimedean absolute value of any eigenvalue of $T_m(p)$ and $T_m(p^2,j)$ with $0 \le j \le g$ with right proofs. We write the estimate for any eigenvalue of $T_m(p)$ in a form of $p$-product expansion. We show existence of a Siegel cusp eigenform in every non-zero $\mathfrak {S}_{m}({\mathcal {N}},\chi )$ (for which see Chapter 6 of this paper) $\subset \mathcal {S}_{m}(\Gamma _{g}({\mathcal {N}}))$ whose $p$-parameters satisfy $|\alpha _0(p)|=p^{\frac {gm}{2}-\frac {g(g+1)}{4}}$ and $|\alpha _j(p)|=p^{j-\frac {g+1}{2}}$ for any $1 \le j \le g$ and any prime $p \nmid \mathcal {N}$. Now fix any integer $g \ge 1.$ Let $\{k_{j}| 1\le j\le g \}$ be arbitrary numbers in $2^{-1}{\mathbb {Z}}$ with $\sum ^{g}_{j=1}k_{j}=0$, $0\le k_{j}\le j-\frac {g+1}{2}$ for any $\frac {g+1}{2} \le j \le g$, and $0 \ge k_{j}\ge j - \frac {g+1}{2}$ for any $1\le j<\frac {g+1}{2}$. We show there are infinitely many integers $m \ge g+1$ such that there exists an eigenform $\in \mathcal {S}_{m}(\Gamma _{g}({\mathcal {N}}))$ whose $p$-parameters $\{\alpha _{j}(p) \ | \ 0\le j\le g \}$ satisfy $|\alpha _{0}(p)|=p^{\frac {gm}{2}-\frac {g(g+1)}{4}}$ and $|\alpha _{j}(p)|=p^{k_{j}}$ for any prime $p \nmid \mathcal {N}$ and any integer $j \in [1,g].$ We can let $k_j=0$ for all $j \in [1,g].$ Therefore for any integer $g \ge 1$ we have infinitely many holomorphic Siegel cusp eigenforms of degree $g$ that satisfy the generalized Ramanujan conjecture. For any eigenform$\in \mathcal {S}_m(\Gamma _2(\mathcal {N}))$ with $m \ge 3,$ we give also estimations of $|\alpha _2(p)|$ and $|\alpha _1(p)|$ for any prime $p \nmid \mathcal {N}.$

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