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Pantchichkine, A. . (2023). ALGEBRAIC DIFFERENTIAL OPERATORS ON ARITHMETIC AUTOMORPHIC FORMS, MODULAR DISTRIBUTIONS, 𝙋-ADIC INTERPOLATION OF THEIR CRITICAL 𝙇 VALUES VIA BGG MODULES AND HECKE ALGEBRAS. Thang Long Journal of Science: Mathematics and Mathematical Sciences, 1(4), 1-26. https://science.thanglong.edu.vn/index.php/volc/article/view/39
ALGEBRAIC DIFFERENTIAL OPERATORS ON ARITHMETIC AUTOMORPHIC FORMS, MODULAR DISTRIBUTIONS, 𝙋-ADIC INTERPOLATION OF THEIR CRITICAL 𝙇 VALUES VIA BGG MODULES AND HECKE ALGEBRAS
Main Article Content
Abstract
The paper extends author’s method of modular distributions (2002, [75]) to arithmetic automorphic L functions on general classical groups. Main resultat gives a p-adic interpolation of their critical L values in the form of integrals of distributions constructed from a given eigen function of Hecke algebras by applying BGG modules, (see also preprints [78] and [79].
In particular, algebraic differential operators are described acting on auto-morphic forms ' on unitary groups U(n; n) over an imaginary quadratic field
Applications are given to Shimura’s zeta functions [90] attached special L-values attached to . and normalized in accordance with Deligne’s Gamma factors rule [21]. An explicit description of Shimura’s -factors is used.
Article Details
Keywords
Automorphic forms, classical groups, p-adic L-functions, differential operators, non-archimedean weight spaces, quasi-modular forms, Fourier coefficients
References
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[8] Böcherer, S., and Schmidt, C.-G., p-adic measures attached to Siegel modular forms, Ann. Inst. Fourier 50, N 5, 1375-1443 (2000).
[9] Berthelot, Pierre, Ogus, Arthur, Notes on crystaline cohomology Princeton University Press, 1978.
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forms. Algebra and Number Theory 8:10 (2014)
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[34] Grothendieck, A. On the de Rham cohomology of algebraic varieties Publ. Math. IHES , 29 (1966) pp. 351-359
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[37] Harris, M., Arithmetic vector bundles and automorphic forms on Shimura varieties, II, Comp. Math. 60 (1986), 323-378.
[38] Harris, M., L-functions and periods of polarized regular motives. J. Reine Angew. Math, (483):75-161, 1997.
[39] Harris, M., Automorphic Galois representations and the cohomology of Shimura varieties. Proceedings of the International Congress of Mathe- maticians, Seoul, 2014
[40] Harris M., Labesse, J.-P., Conditional base change for unitary groups, Asian J. Math. 8:4 (2004), 653-683.
[41] Hurwitz, A., Über die Entwicklungskoeffizienten der lemniskatischen Funktionen, Math. Ann., 51 (1899), 196-226; Mathematische Werke. Vols. 1 and 2, Birkhaeuser, Basel, 1962-1963, see Vol. 2, No. LXVII.
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[43] Ichikawa, T., Vector-valued p-adic Siegel modular forms, J. reine angew. Math., DOI 10.1515/ crelle-2012-0066.
[44] Ichikawa, Takashi, Integrality of nearly (holomorphic) Siegel modular forms, arXiv: 1508.03138v2.
[45] Ichikawa, T., Algebraic theory of nearly holomorphic Siegel modular forms, RIMS Kôkyûroku, (2017) No.2036, 31-44. Workshop "Automorphic Forms, Automorphic L-Functions and Related Topics" 2016/02/01-05, Ed.
Shuichi Hayashida).
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[50] Klosin ,K., Maass spaces on U(2,2) and the Bloch-Kato conjecture for the symmetric square motive of a modular form, Journal of the Mathematical Society of Japan, Vol. 67, No. 2 (2015) pp. 797-860.
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[3] Hiraku Atobe, Hisashi Kojima. On the Miyawaki lifts of Hermitian modular forms. Journal of Number Theory 185 (2018) 281-18
[4] Bhatt B., Scholze P., Prisms and Prismatic Cohomology arXiv:1905.08229 [math.AG], latest version 27 Aug 2019 (v2))
[5] [Boe85] Böcherer, S., Über die Funktionalgleichung automorpher L– Funktionen zur Siegelscher Modulgruppe. J. reine angew. Math. 362 (1985) 146-168
[6] Boecherer, S., Nagaoka, S. , On p-adic properties of Siegel modular forms, in: Automorphic Forms. Research in Number Theory from Oman. Springer Proceedings in Mathematics and Statistics 115. Springer 2014.
[7] Böcherer, S., Panchishkin, A.A., Higher Twists and Higher Gauss Sums Vietnam Journal of Mathematics 39:3 (2011) 309-326
[8] Böcherer, S., and Schmidt, C.-G., p-adic measures attached to Siegel modular forms, Ann. Inst. Fourier 50, N 5, 1375-1443 (2000).
[9] Berthelot, Pierre, Ogus, Arthur, Notes on crystaline cohomology Princeton University Press, 1978.
[10] Bouganis T., Non-abelian p-adic L-functions and Eisenstein series of unitary groups; the CM method, Ann. Inst. Fourier (Grenoble), 64 no. 2 (2014), p. 793-891.
[11] Bouganis T., p-adic Measures for Hermitian Modular Forms and the Rankin-Selberg Method. in Elliptic Curves, Modular Forms and Iwasawa Theory - Conference in honour of the 70th birthday of John Coates, pp 33-86
[12] Brinon, Olivier and Conrad, Brian CMI Summer School Notes on p-Adic Hodge Theory, 2009
[13] Braun, H. Hermitian modular functions. III, Ann. of Math. (2) 53 (1951), 143-160.
[14] Caraiani A., Eischen E., Fintzen J., Mantovan E., Varma I., p-adic q-expansion principles on unitary Shimura varieties, Directions in number theory, vol. 3, Springer, 2016, pp. 197-243.
[15] Clozel, L., Motifs et formes automorphes: Applications du principe de fonctorialité, pp. 77-159 in Automorphic formos, Shimura varieties, and L-functions (Ann Arbor, MI, 1988), vol. 1, edited by L. Clozel and J. S. Milne, Perspectives in Mathematics 10, Academic Press, Boston, MA, 1990.
[16] Courtieu, M., Panchishkin, A.A., Non-Archimedean L-Functions and Arithmetical Siegel Modular Forms, Lecture Notes in Mathematics 1471, Springer-Verlag, 2004 (2nd augmented ed.)
[17] Coates, J., Perrin-Riou, B., On p-adic L-functions Attached to Mo-tives over Q. Advanced Studies in Pure Mathematics 17, 1989 Algebraic Number Theory in honor of K. Iwasawa pp. 23-54
[18] Coates, J., On p-adic L-functions Attached to Motives over Q, II. Bo-letim da Sociedade Brasileira de Matemática - Bulletin/Brazilian Mathe- matical Society October 1989, Volume 20, Issue 1, pp 101–112
[19] Coates, J. and Wiles, A., On the conjecture of Birch and Swinnerton-Dyer, Inventiones math. 39, 223-251
[20] Cohen, H. Computing L -Functions: A Survey. Journal de théorie des nombres de Bordeaux, Tome 27 (2015) no. 3 , p. 699-726
[21] Deligne P., Valeurs de fonctions L et périodes d’intégrales, Proc.Sympos.Pure Math. vol. 55. Amer. Math. Soc., Providence, RI, 1979 , 313-346.
[22] Lassina Dembélé, Alexei Panchishkin, John Voight, and Wadim Zudilin, Special hypergeometric motives and their L-functions: Asai recognition arXiv:1906.07384v2 [math.NT]
[23] Do, Anh Tuan, p-Adic Admissible Measures Attached to Siegel Modular Forms of Arbitrary Genus. Vietnam Journal of Mathematics December 2017, Volume 45, Issue 4, pp 695–711
[24] Eischen, Ellen E., p-Adic Differential Operators on Automorphic Forms on Unitary Groups. Annales de l’Institut Fourier 62, No.1 (2012) 177-243. [EE14] Eischen, Ellen E., Eisenstein measure for vector-weight automorphic
forms. Algebra and Number Theory 8:10 (2014)
[25] Eischen Ellen E., Harris, Michael, Li, Jian-Shu, Skinner, Christo-pher M., p-adic L-functions for unitary groups, arXiv:1602.01776v4 [math.NT] (Mon, 22 Jul 2019)
[26] Eichler, M., Zagier, D., The theory of Jacobi forms, Progress in Math- ematics, vol. 55 (Birkhäuser, Boston, MA, 1985). [Ike01] Ikeda, T., On the lifting of elliptic cusp forms to Siegel cusp forms of degree 2n, Ann. of Math. (2) 154 (2001), 641-681.
[27] Ikeda, T., On the lifting of Hermitian modular forms, Compositio Math. 144, 1107-1154, (2008)
[28] Iwasawa, K., Lectures on p-Adic L-Functions, Ann. of Math. Studies, N 74. Princeton Univ. Press (1972).
[29] Paul Garrett , Universality of Holomorphic Dis-crete Series. (February 19, 2005) garrett@math.umn.edu http://www.math.umn.edu/˜garrett/
[30] [GMPS14] Gelbart, S., Miller, S.D, Panchishkin, S., and Shahidi, F., A p-adic integral for the reciprocal of L-functions. Travaux du Col- loque "Automorphic Forms and Related Geometry, Assessing the Legacy of I.I. Piatetski-Shapiro" (23 - 27 April, 2012, Yale University in New Haven, CT), Contemporary Mathematics, 345-374 (avec Stephen Gelbart, Stephen D. Miller, and Freydoon Shahidi), 53-68, 2014.
[31] Gelbart, S., and Shahidi, F., Analytic Properties of Automorphic L-functions, Academic Press, New York, 1988.
[32] Gelbart S.,Piatetski-Shapiro I.I., Rallis S. Explicit constructions of automorphic L-functions. Springer-Verlag, Lect. Notes in Math. N 1254 (1987) 152p.
[33] Gritsenko, V.A ., Zeta function of degree six of Hermitian modular forms of genus 2, J. Soviet Math. 43 (1988), 2540-2553.
[34] Grothendieck, A. On the de Rham cohomology of algebraic varieties Publ. Math. IHES , 29 (1966) pp. 351-359
[35] Harris, M., Special values of zeta functions attached to Siegel modular forms. Ann. Sci. Ecole Norm Sup. 14 (1981), 77-120.
[36] Harris, M., Arithmetic vector bundles and automorphic forms on Shimura varieties, I, Invent. Math. 82 (1985), 151-189.
[37] Harris, M., Arithmetic vector bundles and automorphic forms on Shimura varieties, II, Comp. Math. 60 (1986), 323-378.
[38] Harris, M., L-functions and periods of polarized regular motives. J. Reine Angew. Math, (483):75-161, 1997.
[39] Harris, M., Automorphic Galois representations and the cohomology of Shimura varieties. Proceedings of the International Congress of Mathe- maticians, Seoul, 2014
[40] Harris M., Labesse, J.-P., Conditional base change for unitary groups, Asian J. Math. 8:4 (2004), 653-683.
[41] Hurwitz, A., Über die Entwicklungskoeffizienten der lemniskatischen Funktionen, Math. Ann., 51 (1899), 196-226; Mathematische Werke. Vols. 1 and 2, Birkhaeuser, Basel, 1962-1963, see Vol. 2, No. LXVII.
[42] Hua, L.K. Harmonic analysis of functions of several complex variables in the classical domains, Transl. Math. Monographs 6, AMS 1963.
[43] Ichikawa, T., Vector-valued p-adic Siegel modular forms, J. reine angew. Math., DOI 10.1515/ crelle-2012-0066.
[44] Ichikawa, Takashi, Integrality of nearly (holomorphic) Siegel modular forms, arXiv: 1508.03138v2.
[45] Ichikawa, T., Algebraic theory of nearly holomorphic Siegel modular forms, RIMS Kôkyûroku, (2017) No.2036, 31-44. Workshop "Automorphic Forms, Automorphic L-Functions and Related Topics" 2016/02/01-05, Ed.
Shuichi Hayashida).
[46] Katz, N.M., p-adic interpolation of real analytic Eisenstein series. Ann. of Math. 104 (1976) 459–571
[47] Katz, N.M., p- adic L-functions for CM-fields. Invent. Math. 48 (1978) 199-297
[48] Katz, N. M., Oda, T., On the differentiation of de Rham cohomology classes with respect to parameters. J. Math. Kyoto Univ. 8 1968 199-213.
[49] Kikuta, Toshiyuki, Nagaoka, Shoyu, Note on mod p property of Hermi-tian modular forms arXiv:1601.03506 [math.NT]
[50] Klosin ,K., Maass spaces on U(2,2) and the Bloch-Kato conjecture for the symmetric square motive of a modular form, Journal of the Mathematical Society of Japan, Vol. 67, No. 2 (2015) pp. 797-860.
[51] [Ko80] Koblitz, Neal, p-adic Analysis. A Short Course on Recent Work, Cambridge Univ. Press, 1980 [52] Kubota, T., Leopoldt, H.–W. (1964): Eine p�adische Theorie der Zetawerte. I. J. reine u. angew. Math., 214/215, 328-339 (1964).
[60] Kai-Wen Lan, Arithmetic compactifications of PEL-type shimura varieties, London Mathematical Society Monographs, vol. 36, Princeton University Press, 2013.
[61] Lang, Serge. Introduction to modular forms. With appendixes by D. Zagier and Walter Feit. Springer-Verlag, Berlin, 1995
[62] Manin, Yu. I., Periods of cusp forms and p-adic Hecke series, Mat. Sbornik, 92 , 1973, pp. 378-401
[63] Manin, Yu. I., Non-Archimedean integration and Jacquet-Langlands p-adic L-functions, Uspekhi Mat. Nauk, 1976, Volume 31, Issue 1(187), 5-54
[64] Manin, Yu. I.,Panchishkin, A.A., Introduction to Modern Number The- ory: Fundamental Problems, Ideas and Theories (Encyclopaedia of Math-ematical Sciences), Second Edition, 504 p., Springer (2005)
[65] Manin, Yu.I., Vishik, M. M., p-adic Hecke series of imaginary quadratic fields, (Russian) Mat. Sb. (N.S.) 95(137) (1974), 357-383.
[66] Martin, François , Royer, Emmanuel , Formes modulaires et péri-odes. Formes modulaires et transcendance, 1-117, Sémin. Congr., 12, Soc. Math. France, Paris (2005).
[67] Martin, François , Royer, Emmanuel , Rankin-Cohen brackets on quasimodular forms J. Ramanujan Math. Soc. 24, No.3 (2009) 213-233
[68] Mazur, B., Tate J., Teitelbaum, J., On p-adic analogues of the con- jectures of Birch and Swinnerton-Dyer. Invent. Math. 84, 1-48 (1986).
[69] Milnor, J., Stasheff, J., Characteristic Classes, Ann. of Math. Studies N 76, Princeton Univ. Press. (1974), p 231-264.
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