With cedram.org English français Home Overview Search for an article Submit a paper Informations for the authors Subscription Limited access - RUCHE Table of contents for this issue | Previous article | Next article Ellen E. Eischen$p$-adic Differential Operators on Automorphic Forms on Unitary Groups(Opérateurs différentiels $p$-adiques sur formes automorphes pour groupes unitaires)Annales de l'institut Fourier, 62 no. 1 (2012), p. 177-243, doi: 10.5802/aif.2704 Article PDF | Reviews MR 2986270 | Zbl 1257.11054 Class. Math.: 14G35, 11G10, 11F03, 11F55, 11F60Keywords: $p$-adic automorphic forms, differential operators, Maass operators Résumé - AbstractThe 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 $U(n)\times U(n)$. Bibliography[1] Michel Courtieu & Alexei Panchishkin, Non-Archimedean $L$-functions and arithmetical Siegel modular forms, Lecture Notes in Mathematics 1471, Springer-Verlag, Berlin, 2004  MR 2034949[2] Ellen Eischen, $p$-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] Ellen E. Eischen, “An Eisenstein Measure for Unitary Groups”, , In Preparation. [4] Ellen E. Eischen, Michael Harris, Jian-Shu Li & Christopher M. Skinner, “$p$-adic $L$-functions for Unitary Shimura Varieties, II”, , In preparation. [5] Gerd Faltings & Ching-Li Chai, 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 1083353 |  Zbl 0744.14031[6] Michael Harris, “Special values of zeta functions attached to Siegel modular forms”, Ann. Sci. École Norm. Sup. (4) 14 (1981) no. 1, p. 77-120 Numdam |  MR 618732 |  Zbl 0465.10022[7] Michael Harris, “Arithmetic vector bundles and automorphic forms on Shimura varieties. II”, Compositio Math. 60 (1986) no. 3, p. 323-378 Numdam |  MR 869106 |  Zbl 0612.14019[8] Michael Harris, Jian-Shu Li & Christopher M. Skinner, “$p$-adic $L$-functions for unitary Shimura varieties. I. Construction of the Eisenstein measure”, Doc. Math. (2006) no. Extra Vol., p. 393-464 (electronic)  MR 2290594[9] Haruzo Hida, $p$-adic automorphic forms on Shimura varieties, Springer Monographs in Mathematics, Springer-Verlag, New York, 2004  MR 2055355[10] Haruzo Hida, “$p$-adic automorphic forms on reductive groups”, Astérisque (2005) no. 298, p. 147-254, Automorphic forms. I  MR 2141703[11] , Notes on Nicholas Katz’s lectures in the seminar on the Sato-Tate Conjecture at Princeton University during the fall of 2006 [12] Nicholas Katz, Travaux de Dwork, Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 409, Springer, 1973, p. 167–200. Lecture Notes in Math., Vol. 317 Numdam |  MR 498577 |  Zbl 0259.14007[13] Nicholas M. Katz, “Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin”, Inst. Hautes Études Sci. Publ. Math. (1970) no. 39, p. 175-232 Numdam |  MR 291177 |  Zbl 0221.14007[14] Nicholas M. Katz, $p$-adic properties of modular schemes and modular forms, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Springer, 1973, p. 69–190. Lecture Notes in Mathematics, Vol. 350  MR 447119 |  Zbl 0271.10033[15] Nicholas M. Katz, “The Eisenstein measure and $p$-adic interpolation”, Amer. J. Math. 99 (1977) no. 2, p. 238-311 Article |  MR 485797 |  Zbl 0375.12022[16] Nicholas M. Katz, “$p$-adic $L$-functions for CM fields”, Invent. Math. 49 (1978) no. 3, p. 199-297 Article |  MR 513095 |  Zbl 0417.12003[17] Nicholas M. Katz & Tadao Oda, “On the differentiation of de Rham cohomology classes with respect to parameters”, J. Math. Kyoto Univ. 8 (1968), p. 199-213 Article |  MR 237510 |  Zbl 0165.54802[18] Kiran Kedlaya, $p$-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 2482348[19] Robert E. Kottwitz, “Points on some Shimura varieties over finite fields”, J. Amer. Math. Soc. 5 (1992) no. 2, p. 373-444 Article |  MR 1124982 |  Zbl 0796.14014[20] Kai-Wen Lan, 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] Hans Maass, Differentialgleichungen und automorphe Funktionen, in Proceedings of the International Congress of Mathematicians, 1954, Amsterdam, vol. III, Erven P. Noordhoff N.V., Groningen, 1956, p. 34-39  MR 86901 |  Zbl 0074.30402[22] Hans Maass, 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 344198 |  Zbl 0224.10028[23] James Milne, Introduction to Shimura Varieties, 2004, Notes available at http://www.jmilne.org/math/ [24] David Mumford, “An analytic construction of degenerating abelian varieties over complete rings”, Compositio Math. 24 (1972), p. 239-272 Numdam |  MR 352106 |  Zbl 0241.14020[25] A. A. Panchishkin, “Two variable $p$-adic $L$-functions attached to eigenfamilies of positive slope”, Invent. Math. 154 (2003) no. 3, p. 551-615 Article |  MR 2018785[26] A. A. Panchishkin, “The Maass-Shimura differential operators and congruences between arithmetical Siegel modular forms”, Mosc. Math. J. 5 (2005) no. 4, p. 883-918, 973–974  MR 2266464[27] M. Rapoport, “Compactifications de l’espace de modules de Hilbert-Blumenthal”, Compositio Math. 36 (1978) no. 3, p. 255-335 Numdam |  MR 515050 |  Zbl 0386.14006[28] Jean-Pierre Serre, Formes modulaires et fonctions zêta $p$-adiques, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, 1972), Springer, 1973, p. 191–268. Lecture Notes in Math., Vol. 350  MR 404145 |  Zbl 0277.12014[29] Goro Shimura, “Arithmetic of differential operators on symmetric domains”, Duke Math. J. 48 (1981) no. 4, p. 813-843 Article |  MR 782579 |  Zbl 0487.10021[30] Goro Shimura, “Differential operators and the singular values of Eisenstein series”, Duke Math. J. 51 (1984) no. 2, p. 261-329 Article |  MR 747868 |  Zbl 0546.10025[31] Goro Shimura, “Invariant differential operators on Hermitian symmetric spaces”, Ann. of Math. (2) 132 (1990) no. 2, p. 237-272 Article |  MR 1070598 |  Zbl 0718.11020[32] Goro Shimura, “Differential operators, holomorphic projection, and singular forms”, Duke Math. J. 76 (1994) no. 1, p. 141-173 Article |  MR 1301189 |  Zbl 0829.11029[33] Goro Shimura, Abelian varieties with complex multiplication and modular functions, Princeton Mathematical Series 46, Princeton University Press, Princeton, NJ, 1998  MR 1492449 |  Zbl 0908.11023[34] Goro Shimura, Arithmeticity in the theory of automorphic forms, Mathematical Surveys and Monographs 82, American Mathematical Society, Providence, RI, 2000  MR 1780262 © Annales de L'Institut Fourier - ISSN (électronique) : 1777-5310