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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
Table of Contents
Chapters
- 1. Introduction
- 2. Action of Double Cosets
- 3. Satakeâs 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. Langlandsâ 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 $p \nmid \mathcal {N}.$- Anatolij N. Andrianov, Quadratic forms and Hecke operators, Springer, Berlin, 1987.
- Don Blasius and Jonathan D. Rogawski, Zeta functions of Shimura varieties, Proc. Sympos. Pure Math., vol. 55, Part 2, Amer. Math. Soc., Providence, RI, 1994, pp. 525â571.
- A. Borel and H. Jacquet, Automorphic forms and automorphic representations, Proc. Sympos. Pure Math., XXXIII, Part 1, Amer. Math. Soc., Providence, R.I., 1979, pp. 189â202.
- A. Borel, Automorphic $L$-functions, Proc. Sympos. Pure Math., XXXIII, Part 2, Amer. Math. Soc., Providence, R.I., 1979, pp. 27â61.
- S. Bosch, U. GĂŒntzer, and R. Remmert, Non-Archimedean analysis, Springer-Verlag, Berlin, 1984.
- Valentin Blomer and Farrell Brumley, The role of the Ramanujan conjecture in analytic number theory, Bull. Amer. Math. Soc. (N.S.) 50 (2013), 267â320.
- Ching-Li Chai, Compactification of Siegel moduli schemes, Cambridge University Press, Cambridge, 1985.
- Pierre Deligne, Formes modulaires et reprĂ©sentations $l$-adiques, Lecture Notes in Math., vol. 179, Springer, Berlin, 1971, pp. 139â172.
- Pierre Deligne, La conjecture de Weil. I and II, Inst. Hautes Ătudes Sci. Publ. Math. 43 and 52 (1971 and 1980), 273â307 and 137â252.
- Pierre Deligne and Jean-Pierre Serre, Formes modulaires de poids $1$, Ann. Sci. Ăcole Norm. Sup. (4) 7 (1974), 507â530.
- Gerd Faltings, $p$-adic Hodge theory, J. Amer. Math. Soc. 1 (1988), no. 1, 255â299.
- Gerd Faltings, Crystalline cohomology and $p$-adic Galois-representations, in âAlgebraic Analysis, Geometry and Number Theoryâ, Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 25â80.
- Gerd Faltings, $F$-isocrystals on open varieties: results and conjectures, Progr. Math., vol. 87, BirkhĂ€user Boston, Boston, MA, 1990, pp. 219â248.
- Gerd Faltings and Ching-Li Chai, Degeneration of abelian varieties, Springer-Verlag, Berlin, 1990.
- E. Freitag, Siegelsche Modulfunktionen, Springer-Verlag, Berlin, 1983.
- Stephen S. Gelbart, Automorphic forms on adĂšle groups, Princeton University Press, Princeton, N.J., 1975.
- Stephen Gelbart, Three lectures on the modularity of $\overline \rho _{E,3}$ and the Langlands reciprocity conjecture, in âModular forms and Fermatâs last theoremâ, Springer, New York, 1997, pp. 155â207.
- Kazuyuki Hatada, Siegel cusp forms as holomorphic differential forms on certain compact varieties, Math. Ann. 262 (1983), 503â509.
- Kazuyuki Hatada, Homology groups, differential forms and Hecke rings on Siegel modular varieties, in âTopics in Mathematical Analysisâ, World Sci. Pub., Singapore, New Jersey, London, 1989, pp. 371â409. (Corrections of several misprints in this paper are published in the appendix of K. Hatada, On the local zeta functions of compactified Hilbert modular schemes and action of the Hecke rings, Sci. Rep. Fac. Educ., Gifu Univ. (Nat. Sci.) Vol. 18(2), (1994), 1â34).
- Kazuyuki Hatada, Correspondences for Hecke rings and $l$-adic cohomology groups on smooth compactifications of Siegel modular varieties, Proc. Japan Acad. Ser. A Math. Sci. 65 (1989), no. 2, 62â65.
- Kazuyuki Hatada, On the action of Hecke rings on homology groups of smooth compactifications of Siegel modular varieties and Siegel cusp forms, Tokyo J. Math. 13 (1990), no. 1, 191â205.
- Kazuyuki Hatada, Estimates for eigenvalues of Hecke operators on Siegel cusp forms, J. Reine Angew. Math. 480 (1996), 105â123.
- Kazuyuki Hatada, On classical and $l$-adic modular forms of levels $Nl^m$ and $N$, J. Number Theory 87 (2001), no. 1, 1â14.
- Heisuke Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, I and II, Ann. of Math. 79 (1964), 109â203 and 205â326.
- Roland Huber, Ătale cohomology of rigid analytic varieties and adic spaces, Friedr. Vieweg & Sohn, Braunschweig, 1996.
- Nicholas M. Katz and William Messing, Some consequences of the Riemann hypothesis for varieties over finite fields, Invent. Math. 23 (1974), 73â77.
- A. W. Knapp, Introduction to the Langlands program, Proc. Sympos. Pure Math., vol. 61, Amer. Math. Soc., Providence, RI, 1997, pp. 245â302.
- R. P. Langlands, On the notion of an automorphic representation, A supplement to the preceding paper, Proc. Sympos. Pure Math., vol. XXXIII, Part 1, Amer. Math. Soc., Providence, R.I., 1979, pp. 203â207.
- Henry B. Laufer, On rational singularities, Amer. J. Math. 94 (1972), 597â608.
- James S. Milne, Ătale cohomology, Princeton Mathematical Series, vol. 33, Princeton University Press, Princeton, N.J., 1980.
- Yukihiko Namikawa, Toroidal compactification of Siegel spaces, Lecture Notes in Math., vol. 812, Springer, Berlin, 1980.
- Alexey A. Panchishkin, Non-Archimedean $L$-functions of Siegel and Hilbert modular forms, Lecture Notes in Math., vol. 1471, Springer, Berlin, 1991.
- Michel Raynaud, GĂ©omĂ©trie analytique rigide dâaprĂšs Tate, Kiehl,$\cdots$ Table ronde dâanalyse non archimĂ©dienne, Bull. Soc. Math. France 39/40 (1974), 319â327.
- IchirĂŽ Satake, Spherical functions and Ramanujan conjecture, Proc. Sympos. Pure Math., IX, Amer. Math. Soc., Providence, R.I., 1966, pp. 258â264.
- Jean-Pierre Serre, Homologie singuliĂšre des espaces fibrĂ©s. Applications, Ann. of Math. 54 (1951), 425â505.
- J.-P. Serre, Sur la topologie des variĂ©tĂ©s algĂ©briques en caractĂ©ristique $p$, in âCollected Papersâ Vol. I (1986), Springer, Berlin Heidelberg, 501â530.
- Jean-Pierre Serre, Corps locaux, Hermann, Paris, 1968.
- Jean-Pierre Serre, Abelian $l$-adic representations and elliptic curves, W. A. Benjamin, Inc., New York-Amsterdam, 1968.
- J.-P. Serre, Une interprĂ©tation des congruences relatives Ă la fonction $\tau$ de Ramanujan, SĂ©minaire Delange-Pisot-Poitou 1967/68, $\text {n}^\circ$ 14 in âCollected Papersâ Vol. II (1986), Springer, Berlin Heidelberg, 498â511.
- J.-P. Serre, A course in arithmetic, Springer, New York-Heidelberg, 1973.
- Jean-Pierre Serre, Valeurs propres des endomorphismes de Frobenius (dâaprĂšs P. Deligne), SĂ©minaire Bourbaki 1973/1974, no 446 in âCollected Papersâ Vol. III (1986), Springer, Berlin, pp. 179â188.
- J.-P. Serre, Modular forms of weight one and Galois representations, in âAlgebraic number fieldsâ, Academic Press, London, 1977, pp. 193â268. (âCollected Papers,â Vol. III (1986), Springer, Berlin Heidelberg, 292â367).
- Jean-Pierre Serre, Groupes algĂ©briques associĂ©s aux modules de Hodge-Tate, AstĂ©risque, vol. 65, Soc. Math. France, Paris, 1979, pp. 155â188.
- Jean-Pierre Serre, Sur les reprĂ©sentations modulaires de degrĂ© $2$ de $\textrm {Gal}(\overline \textbf {Q}/\textbf {Q})$, Duke Math. J. 54 (1987), no. 1, 179â230.
- Jean-Pierre Serre, Lectures on $N_X (p)$, CRC Press, Boca Raton, FL, 2012.
- SGA 4 Théorie des topos et cohomologie étale des schémas. Tome 1, 2, 3, Lecture Notes in Mathematics, Vols. 269, 270, 305, Springer, Berlin-New York, 1972, 1973.
- Goro Shimura, On modular correspondences for $Sp(n,\,Z)$ and their congruence relations, Proc. Nat. Acad. Sci. U.S.A. 49 (1963), 824â828.
- Goro Shimura, Algebraic number fields and symplectic discontinuous groups, Ann. of Math. 86 (1967), 503â592.
- Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Tokyo; Princeton University Press, Princeton, N.J., 1971.
- Goro Shimura, On the Fourier coefficients of modular forms of several variables, Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. II 17 (1975), 261â268.
- Tetsuji Shioda, On elliptic modular surfaces, J. Math. Soc. Japan 24 (1972), 20â59.
- Richard Taylor, On the $l$-adic cohomology of Siegel threefolds, Invent. Math. 114 (1993), no. 2, 289â310.
- Rainer Weissauer, Endoscopy for $\textrm {GSp}(4)$ and the cohomology of Siegel modular threefolds, Vol. 1968, Springer-Verlag, Berlin, 2009.