$p$-adic Differential Operators on Automorphic Forms on Unitary Groups

Eischen, Ellen E.

Annales de l'Institut Fourier, Tome 62 (2012) no. 1, pp. 177-243.

Nous construisons certains opérateurs différentiels $C^{\infty }$ et leurs analogues $p$-adiques, qui agissent sur des formes automorphes (à valeurs vectorielles ou scalaires) pour les groupes unitaires $U(n,n)$. Nous étudions des propriétés de ces opérateurs, et nous les utilisons à prouver quelques théorèmes arithmetiques. Ces opérateurs différentiels sont une généralisation au cas $p$-adique des opérateurs différentiels $C^{\infty }$ étudiés d’abord par H. Maass et étudiés ensuite en détail par M. Harris et G. Shimura. Ils sont une généralisation au cas des opérateurs différentiels $p$-adiques à valeurs vectorielles construits pour les formes modulaires par N. Katz. Ils devraient être utiles dans la construction de certaines fonctions $L$ $p$-adiques, en particulier les fonctions $L$ $p$-adiques attachées aux familles $p$-adiques de formes automorphes pour les groupes unitaires $U\left(n\right)\times U\left(n\right)$.

The goal of this paper is to study certain $p$-adic differential operators on automorphic forms on $U(n,n)$. These operators are a generalization to the higher-dimensional, vector-valued situation of the $p$-adic differential operators constructed for Hilbert modular forms by N. Katz. They are a generalization to the $p$-adic case of the $C^{\infty }$-differential operators first studied by H. Maass and later studied extensively by M. Harris and G. Shimura. The operators should be useful in the construction of certain $p$-adic $L$-functions attached to $p$-adic families of automorphic forms on the unitary groups $U\left(n\right)\times U\left(n\right)$.

Classification : 14G35, 11G10, 11F03, 11F55, 11F60
Mots clés : $p$-adic automorphic forms, differential operators, Maass operators

[1] Courtieu, Michel; Panchishkin, Alexei Non-Archimedean $L$-functions and arithmetical Siegel modular forms, Lecture Notes in Mathematics, 1471, Springer-Verlag, Berlin, 2004 | MR

[2] Eischen, Ellen $p$-adic differential operators on vector-valued automorphic forms and applications (2009) (Ph.D. thesis, University of Michigan, available at http://www.math.northwestern.edu/ eeischen/EischenThesisSubmitted061109.pdf)

[3] Eischen, Ellen E. An Eisenstein Measure for Unitary Groups (In Preparation.)

[4] Eischen, Ellen E.; Harris, Michael; Li, Jian-Shu; Skinner, Christopher M. $p$-adic $L$-functions for Unitary Shimura Varieties, II (In preparation.)

[5] Faltings, Gerd; Chai, Ching-Li Degeneration of abelian varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 22, Springer-Verlag, Berlin, 1990 (With an appendix by David Mumford) | MR | Zbl

[6] Harris, Michael Special values of zeta functions attached to Siegel modular forms, Ann. Sci. École Norm. Sup. (4), Volume 14 (1981) no. 1, pp. 77-120 | Numdam | MR | Zbl

[7] Harris, Michael Arithmetic vector bundles and automorphic forms on Shimura varieties. II, Compositio Math., Volume 60 (1986) no. 3, pp. 323-378 | Numdam | MR | Zbl

[8] Harris, Michael; Li, Jian-Shu; Skinner, Christopher M. $p$-adic $L$-functions for unitary Shimura varieties. I. Construction of the Eisenstein measure, Doc. Math. (2006) no. Extra Vol., p. 393-464 (electronic) | MR

[9] Hida, Haruzo $p$-adic automorphic forms on Shimura varieties, Springer Monographs in Mathematics, Springer-Verlag, New York, 2004 | MR

[10] Hida, Haruzo $p$-adic automorphic forms on reductive groups, Astérisque (2005) no. 298, pp. 147-254 (Automorphic forms. I) | Numdam | MR

[11] (Notes on Nicholas Katz’s lectures in the seminar on the Sato-Tate Conjecture at Princeton University during the fall of 2006)

[12] Katz, Nicholas Travaux de Dwork, Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 409, Springer, Berlin, 1973, p. 167-200. Lecture Notes in Math., Vol. 317 | Numdam | MR | Zbl

[13] Katz, Nicholas M. Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin, Inst. Hautes Études Sci. Publ. Math. (1970) no. 39, pp. 175-232 | DOI | Numdam | MR | Zbl

[14] Katz, Nicholas M. $p$-adic properties of modular schemes and modular forms, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Springer, Berlin, 1973, p. 69-190. Lecture Notes in Mathematics, Vol. 350 | MR | Zbl

[15] Katz, Nicholas M. The Eisenstein measure and $p$-adic interpolation, Amer. J. Math., Volume 99 (1977) no. 2, pp. 238-311 | DOI | MR | Zbl

[16] Katz, Nicholas M. $p$-adic $L$-functions for CM fields, Invent. Math., Volume 49 (1978) no. 3, pp. 199-297 | DOI | MR | Zbl

[17] Katz, Nicholas M.; Oda, Tadao On the differentiation of de Rham cohomology classes with respect to parameters, J. Math. Kyoto Univ., Volume 8 (1968), pp. 199-213 | MR | Zbl

[18] Kedlaya, Kiran $p$-adic cohomology: from theory to practice, $p$-adic Geometry: Lectures from the 2007 Arizona Winter School, American Mathematical Society, 2008, p. 175-200. University Lecture Series, Vol. 45 | MR

[19] Kottwitz, Robert E. Points on some Shimura varieties over finite fields, J. Amer. Math. Soc., Volume 5 (1992) no. 2, pp. 373-444 | DOI | MR | Zbl

[20] Lan, Kai-Wen Arithmetic compactifications of PEL-type Shimura varieties (2008) (Ph.D. thesis, Harvard University, available at http://www.math.princeton.edu/ klan/articles/cpt-PEL-type-thesis-single.pdf)

[21] Maass, Hans Differentialgleichungen und automorphe Funktionen, Proceedings of the International Congress of Mathematicians, 1954, Amsterdam, vol. III, Erven P. Noordhoff N.V., Groningen (1956), pp. 34-39 | MR | Zbl

[22] Maass, Hans Siegel’s modular forms and Dirichlet series, Lecture Notes in Mathematics, Vol. 216, Springer-Verlag, Berlin, 1971 (Dedicated to the last great representative of a passing epoch. Carl Ludwig Siegel on the occasion of his seventy-fifth birthday) | MR | Zbl

[23] Milne, James Introduction to Shimura Varieties (2004) (Notes available at http://www.jmilne.org/math/)

[24] Mumford, David An analytic construction of degenerating abelian varieties over complete rings, Compositio Math., Volume 24 (1972), pp. 239-272 | Numdam | MR | Zbl

[25] Panchishkin, A. A. Two variable $p$-adic $L$-functions attached to eigenfamilies of positive slope, Invent. Math., Volume 154 (2003) no. 3, pp. 551-615 | DOI | MR

[26] Panchishkin, A. A. The Maass-Shimura differential operators and congruences between arithmetical Siegel modular forms, Mosc. Math. J., Volume 5 (2005) no. 4, p. 883-918, 973–974 | MR

[27] Rapoport, M. Compactifications de l’espace de modules de Hilbert-Blumenthal, Compositio Math., Volume 36 (1978) no. 3, pp. 255-335 | Numdam | MR | Zbl

[28] Serre, Jean-Pierre Formes modulaires et fonctions zêta $p$-adiques, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, 1972), Springer, Berlin, 1973, p. 191-268. Lecture Notes in Math., Vol. 350 | MR | Zbl

[29] Shimura, Goro Arithmetic of differential operators on symmetric domains, Duke Math. J., Volume 48 (1981) no. 4, pp. 813-843 | DOI | MR | Zbl

[30] Shimura, Goro Differential operators and the singular values of Eisenstein series, Duke Math. J., Volume 51 (1984) no. 2, pp. 261-329 | DOI | MR | Zbl

[31] Shimura, Goro Invariant differential operators on Hermitian symmetric spaces, Ann. of Math. (2), Volume 132 (1990) no. 2, pp. 237-272 | DOI | MR | Zbl

[32] Shimura, Goro Differential operators, holomorphic projection, and singular forms, Duke Math. J., Volume 76 (1994) no. 1, pp. 141-173 | DOI | MR | Zbl

[33] Shimura, Goro Abelian varieties with complex multiplication and modular functions, Princeton Mathematical Series, 46, Princeton University Press, Princeton, NJ, 1998 | MR | Zbl

[34] Shimura, Goro Arithmeticity in the theory of automorphic forms, Mathematical Surveys and Monographs, 82, American Mathematical Society, Providence, RI, 2000 | MR