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\(p\)-adic differential operators on automorphic forms on unitary groups. (English. French summary) Zbl 1257.11054
This paper studies various aspects of certain \(p\)-adic differential operators on automorphic forms on \(U(n,n)\), which generalize the higher dimensional vector-valued \(p\)-adic differential operators for Hilbert modular forms constructed by N. Katz. These operators also provide the \(p\)-adic version of the \(C^\infty\) differential operators studied first by H. Maass and later by M. Harris and G. Shimura. They should be useful in the construction of certain \(p\)-adic \(L\)-functions attached to \(p\)-adic families of automorphic forms on \(U(n) \times U(n)\).

MSC:
11F85 \(p\)-adic theory, local fields
11F03 Modular and automorphic functions
11F55 Other groups and their modular and automorphic forms (several variables)
11G10 Abelian varieties of dimension \(> 1\)
14G35 Modular and Shimura varieties
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[1] Courtieu, Michel; Panchishkin, Alexei, Non-Archimedean \(L\)-functions and arithmetical Siegel modular forms, 1471, (2004), Springer-Verlag, Berlin · Zbl 1070.11023
[2] Eischen, Ellen \(, p\)-adic differential operators on vector-valued automorphic forms and applications, (2009)
[3] Eischen, Ellen E., An Eisenstein measure for unitary groups · Zbl 1322.11040
[4] Eischen, Ellen E.; Harris, Michael; Li, Jian-Shu; Skinner, Christopher M.\(, p\)-adic \(L\)-functions for unitary Shimura varieties, II · Zbl 1143.11019
[5] Faltings, Gerd; Chai, Ching-Li, Degeneration of abelian varieties, 22, (1990), Springer-Verlag, Berlin · Zbl 0744.14031
[6] Harris, Michael, Special values of zeta functions attached to Siegel modular forms, Ann. Sci. École Norm. Sup. (4), 14, 1, 77-120, (1981) · Zbl 0465.10022
[7] Harris, Michael, Arithmetic vector bundles and automorphic forms on Shimura varieties. II, Compositio Math., 60, 3, 323-378, (1986) · Zbl 0612.14019
[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., Extra Vol., 393-464 (electronic), (2006) · Zbl 1143.11019
[9] Hida, Haruzo \(, p\)-adic automorphic forms on Shimura varieties, (2004), Springer-Verlag, New York · Zbl 1055.11032
[10] Hida, Haruzo \(, p\)-adic automorphic forms on reductive groups, Astérisque, 298, 147-254, (2005) · Zbl 1122.11026
[12] Katz, Nicholas, Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 409, Travaux de dwork, 167-200. Lecture Notes in Math., Vol. 317, (1973), Springer, Berlin · Zbl 0259.14007
[13] Katz, Nicholas M., Nilpotent connections and the monodromy theorem: applications of a result of turrittin, Inst. Hautes Études Sci. Publ. Math., 39, 175-232, (1970) · Zbl 0221.14007
[14] Katz, Nicholas M., Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972)\(, p\)-adic properties of modular schemes and modular forms, 69-190. Lecture Notes in Mathematics, Vol. 350, (1973), Springer, Berlin · Zbl 0271.10033
[15] Katz, Nicholas M., The Eisenstein measure and \(p\)-adic interpolation, Amer. J. Math., 99, 2, 238-311, (1977) · Zbl 0375.12022
[16] Katz, Nicholas M.\(, p\)-adic \(L\)-functions for CM fields, Invent. Math., 49, 3, 199-297, (1978) · Zbl 0417.12003
[17] Katz, Nicholas M.; Oda, Tadao, On the differentiation of de Rham cohomology classes with respect to parameters, J. Math. Kyoto Univ., 8, 199-213, (1968) · Zbl 0165.54802
[18] Kedlaya, Kiran \(, p\)-adic Geometry: Lectures from the 2007 Arizona Winter School \(, p\)-adic cohomology: from theory to practice, 175-200. University Lecture Series, Vol. 45, (2008), American Mathematical Society
[19] Kottwitz, Robert E., Points on some Shimura varieties over finite fields, J. Amer. Math. Soc., 5, 2, 373-444, (1992) · Zbl 0796.14014
[20] Lan, Kai-Wen, Arithmetic compactifications of PEL-type Shimura varieties, (2008) · Zbl 1284.14004
[21] Maass, Hans, Differentialgleichungen und automorphe funktionen, Proceedings of the International Congress of Mathematicians, 1954, Amsterdam, vol. III, 34-39, (1956), Erven P. Noordhoff N.V., Groningen · Zbl 0074.30402
[22] Maass, Hans, Siegel’s modular forms and Dirichlet series, (1971), Springer-Verlag, Berlin · Zbl 0224.10028
[23] Milne, James, Introduction to Shimura Varieties, (2004)
[24] Mumford, David, An analytic construction of degenerating abelian varieties over complete rings, Compositio Math., 24, 239-272, (1972) · Zbl 0241.14020
[25] Panchishkin, A. A., Two variable \(p\)-adic \(L\)-functions attached to eigenfamilies of positive slope, Invent. Math., 154, 3, 551-615, (2003) · Zbl 1065.11025
[26] Panchishkin, A. A., The Maass-Shimura differential operators and congruences between arithmetical Siegel modular forms, Mosc. Math. J., 5, 4, 883-918, 973-974, (2005) · Zbl 1129.11021
[27] Rapoport, M., Compactifications de l’espace de modules de Hilbert-blumenthal, Compositio Math., 36, 3, 255-335, (1978) · Zbl 0386.14006
[28] Serre, Jean-Pierre, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, 1972), Formes modulaires et fonctions zêta \(p\)-adiques, 191-268. Lecture Notes in Math., Vol. 350, (1973), Springer, Berlin · Zbl 0277.12014
[29] Shimura, Goro, Arithmetic of differential operators on symmetric domains, Duke Math. J., 48, 4, 813-843, (1981) · Zbl 0487.10021
[30] Shimura, Goro, Differential operators and the singular values of Eisenstein series, Duke Math. J., 51, 2, 261-329, (1984) · Zbl 0546.10025
[31] Shimura, Goro, Invariant differential operators on Hermitian symmetric spaces, Ann. of Math. (2), 132, 2, 237-272, (1990) · Zbl 0718.11020
[32] Shimura, Goro, Differential operators, holomorphic projection, and singular forms, Duke Math. J., 76, 1, 141-173, (1994) · Zbl 0829.11029
[33] Shimura, Goro, Abelian varieties with complex multiplication and modular functions, 46, (1998), Princeton University Press, Princeton, NJ · Zbl 0908.11023
[34] Shimura, Goro, Arithmeticity in the theory of automorphic forms, 82, (2000), American Mathematical Society, Providence, RI · Zbl 0967.11001
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