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
[1] Patrick B. Allen, Frank Calegari, Ana Caraiani, Toby Gee, David Helm, Bao V. Le Hung, James Newton, Peter Scholze, Richard Taylor, and Jack A. Thorne, Potential automorphy over CM fields, (2018) arXiv:1812.09999 [math.NT]
[2] Amice, Y. and Vélu, J., Distributions p-adiques associées aux series de Hecke, Journées Arithmétiques de Bordeaux (Conf. Univ. Bordeaux, 1974), Astérisque no. 24/25, Soc. Math. France, Paris 1975, pp. 119-131
[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.
[70] My, V. Q. Non-Archimedean Rankin Convolution of Unbounded growth, Math. USSR Sbornik 72 (1992), p 151-161.
[71] Panchishkin, A.A., Non-Archimedean automorphic zeta functions, Moscow University Press (1988).
[72] Panchishkin, A.A., Non-Archimedean L-Functions of Siegel and Hilbert Modular Forms. Volume 1471 (1991)
[73] Panchishkin, A., Motives over totally real fields and p–adic L–functions. Annales de l’Institut Fourier, Grenoble, 44, 4 (1994), 989–1023
[74] Panchishkin, A.A., A new method of constructing p-adic L-functions associated with modular forms, Moscow Mathematical Journal, 2 (2002), Number 2, 1-16
[75] Panchishkin, A. A., Two variable p-adic L functions attached to eigen-families of positive slope, Invent. Math. v. 154, N3 (2003), pp. 551 - 615
[76] Panchishkin, A.A., The Maass–Shimura differential operators and con- gruences between arithmetical Siegel modular forms, Moscow Mathemati-cal Journal, v. 5, N 4, 883-918 (2005).
[77] Panchishkin, A., Analytic constructions of p-adic L-functions and Eisen-stein series. Travaux du Colloque "Automorphic Forms and Related Ge- ometry, Assessing the Legacy of I.I.Piatetski-Shapiro (23-27 April, 2012, Yale University in New Haven, CT)", 345-374, 2014
[78] Panchishkin, A., Algebraic differential operators on unitary groups and their applications. Max-Planck-Institut für Mathe- matik Preprint Series 2021-22 Date of submission: May 27, 2021 https://archive.mpim-bonn.mpg.de/id/eprint/4579/1 /mpim-preprint_2021-22.pdf
[79] Panchishkin, A., New approaches to constructing p-adic L-functions on classical groups, algebraic differential operators, and BGG. Max-Planck-Institut für Mathematik Preprint Series 2021-22, Date of submission: May
27, 2021 https://archive.mpim-bonn.mpg.de/id/eprint/4580/1 /mpim-preprint_2021-23.pdf)
[80] Scholze, P. On torsion in the cohomology of locally symmetric varieties, Annals of Mathematics (2) 182 (2015), no. 3, 945–1066.
[81] Peter Scholze,, p-adic geometry, Proceedings of the ICM 2018.
[82] Shafarevich, I.R. Zeta Function, Moscow University Press (1969).
[83] Sloane N.J.A,, A047817. Denominators of Hurwitz numbers Hn The On-Line Encyclopedia of Integer Sequences https://oeis.org/A047817.
[84] Serre, J.–P., Cours d’arithmétique. Paris, 1970.
[85] Serre, J.–P., Facteurs locaux des fonctions zêta des variétés algéb- riques (définitions et conjectures). Sém. Delange - Pisot - Poitou, exp.19, 1969/70.
[86] Serre, J.–P., Formes modulaires et fonctions zêta p-adiques, Lect Notes in Math. 350 (1973) 191–268 (Springer Verlag)
[87] Shimura G., Introduction to the Arithmetic Theory of Automorphic Func-tions. Princeton: Iwanami Shoten and Princeton Univ. Press; 1971. , Publ.Math. Soc. Japan, No. 11.
[88] Shimura G., Euler Products and Eisenstein series, CBMS Regional Con-ference Series in Mathematics, No.93, Amer. Math. Soc, 1997.
[89] Shimura G., Colloquium Paper: Zeta functions and Eisenstein series on classical groups, Proc Nat. Acad Sci U S A. 1997 Oct 14; 94(21): 11133- 11137
[90] Shimura G., Arithmeticity in the theory of automorphic forms, Mathe-matical Surveys and Monographs, vol. 82 (Amer. Math. Soc., Providence, 2000).
[91] Skinner, Ch, Galois representations associated with unitary groups overQ. Algebra and Number Theory 6:8 (2012)
[92] Skinner, Ch, P-adic L-functions obtained by Eisenstein measure for uni-tary group. https://www.birs.ca/events2018 5-day-workshops 18w5053/videos
[93] Skinner, Ch. and Urban, E. The Iwasawa Main Cconjecture for GL(2). Invent. Math. 195 (2014), no. 1, 1-277. MR 3148103
[94] Urban, E., Nearly Overconvergent Modular Forms, in: Iwasawa Theory 2012. State of the Art and Recent Advances, Contributions in Mathemat- ical and Computational Sciences book series (CMCS, Vol. 7), pp. 401-441
[95] Washington,L., Introduction to Cyclotomic Fields, Springer (1982). [96] Don Zagier, Modular forms and differential operators. Proc. Indian Acad. Sci. (Math. Sci.), 104, No. 1, 1994, pp. 57-75.
[2] Amice, Y. and Vélu, J., Distributions p-adiques associées aux series de Hecke, Journées Arithmétiques de Bordeaux (Conf. Univ. Bordeaux, 1974), Astérisque no. 24/25, Soc. Math. France, Paris 1975, pp. 119-131
[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.
[70] My, V. Q. Non-Archimedean Rankin Convolution of Unbounded growth, Math. USSR Sbornik 72 (1992), p 151-161.
[71] Panchishkin, A.A., Non-Archimedean automorphic zeta functions, Moscow University Press (1988).
[72] Panchishkin, A.A., Non-Archimedean L-Functions of Siegel and Hilbert Modular Forms. Volume 1471 (1991)
[73] Panchishkin, A., Motives over totally real fields and p–adic L–functions. Annales de l’Institut Fourier, Grenoble, 44, 4 (1994), 989–1023
[74] Panchishkin, A.A., A new method of constructing p-adic L-functions associated with modular forms, Moscow Mathematical Journal, 2 (2002), Number 2, 1-16
[75] Panchishkin, A. A., Two variable p-adic L functions attached to eigen-families of positive slope, Invent. Math. v. 154, N3 (2003), pp. 551 - 615
[76] Panchishkin, A.A., The Maass–Shimura differential operators and con- gruences between arithmetical Siegel modular forms, Moscow Mathemati-cal Journal, v. 5, N 4, 883-918 (2005).
[77] Panchishkin, A., Analytic constructions of p-adic L-functions and Eisen-stein series. Travaux du Colloque "Automorphic Forms and Related Ge- ometry, Assessing the Legacy of I.I.Piatetski-Shapiro (23-27 April, 2012, Yale University in New Haven, CT)", 345-374, 2014
[78] Panchishkin, A., Algebraic differential operators on unitary groups and their applications. Max-Planck-Institut für Mathe- matik Preprint Series 2021-22 Date of submission: May 27, 2021 https://archive.mpim-bonn.mpg.de/id/eprint/4579/1 /mpim-preprint_2021-22.pdf
[79] Panchishkin, A., New approaches to constructing p-adic L-functions on classical groups, algebraic differential operators, and BGG. Max-Planck-Institut für Mathematik Preprint Series 2021-22, Date of submission: May
27, 2021 https://archive.mpim-bonn.mpg.de/id/eprint/4580/1 /mpim-preprint_2021-23.pdf)
[80] Scholze, P. On torsion in the cohomology of locally symmetric varieties, Annals of Mathematics (2) 182 (2015), no. 3, 945–1066.
[81] Peter Scholze,, p-adic geometry, Proceedings of the ICM 2018.
[82] Shafarevich, I.R. Zeta Function, Moscow University Press (1969).
[83] Sloane N.J.A,, A047817. Denominators of Hurwitz numbers Hn The On-Line Encyclopedia of Integer Sequences https://oeis.org/A047817.
[84] Serre, J.–P., Cours d’arithmétique. Paris, 1970.
[85] Serre, J.–P., Facteurs locaux des fonctions zêta des variétés algéb- riques (définitions et conjectures). Sém. Delange - Pisot - Poitou, exp.19, 1969/70.
[86] Serre, J.–P., Formes modulaires et fonctions zêta p-adiques, Lect Notes in Math. 350 (1973) 191–268 (Springer Verlag)
[87] Shimura G., Introduction to the Arithmetic Theory of Automorphic Func-tions. Princeton: Iwanami Shoten and Princeton Univ. Press; 1971. , Publ.Math. Soc. Japan, No. 11.
[88] Shimura G., Euler Products and Eisenstein series, CBMS Regional Con-ference Series in Mathematics, No.93, Amer. Math. Soc, 1997.
[89] Shimura G., Colloquium Paper: Zeta functions and Eisenstein series on classical groups, Proc Nat. Acad Sci U S A. 1997 Oct 14; 94(21): 11133- 11137
[90] Shimura G., Arithmeticity in the theory of automorphic forms, Mathe-matical Surveys and Monographs, vol. 82 (Amer. Math. Soc., Providence, 2000).
[91] Skinner, Ch, Galois representations associated with unitary groups overQ. Algebra and Number Theory 6:8 (2012)
[92] Skinner, Ch, P-adic L-functions obtained by Eisenstein measure for uni-tary group. https://www.birs.ca/events2018 5-day-workshops 18w5053/videos
[93] Skinner, Ch. and Urban, E. The Iwasawa Main Cconjecture for GL(2). Invent. Math. 195 (2014), no. 1, 1-277. MR 3148103
[94] Urban, E., Nearly Overconvergent Modular Forms, in: Iwasawa Theory 2012. State of the Art and Recent Advances, Contributions in Mathemat- ical and Computational Sciences book series (CMCS, Vol. 7), pp. 401-441
[95] Washington,L., Introduction to Cyclotomic Fields, Springer (1982). [96] Don Zagier, Modular forms and differential operators. Proc. Indian Acad. Sci. (Math. Sci.), 104, No. 1, 1994, pp. 57-75.