Une nouvelle approche pour construire des fonctions -adiques pour les groupes classiques est présentée comme un projet en cours avec Thanh Hung Dang and Anh Tuan Do (Hanoi, Vietnam). Pour un groupe algébrique sur un corps de nombres les fonctions complexes sont certains produits d’Euler . En particulier, notre construction couvre les fonctions étudiées par Shimura dans [52] via la méthode de doublement de Piatetski-Shapiro et Rallis. Un avatar -adique est une fonction -adique analytique de , interpolant les valeurs spéciales normalisées algébriques de la fonction complexe analytique attachée. Nous utilisons les formes presque-holomorphes et quasi-modulaires générales pour calculer et pour interpoler les valeurs spéciales normalisées.
An approach to constructions of automorphic -functions and their -adic avatars is presented as a work in progress with Thanh Hung Dang and Anh Tuan Do (Hanoi, Vietnam). For an algebraic group over a number field these functions are certain Euler products . In particular, our constructions cover the -functions in [52] via the doubling method of Piatetski-Shapiro and Rallis.
A -adic avatar of is a -adic analytic function of -adic arguments , which interpolates algebraic numbers defined through the normalized critical values of the corresponding complex analytic -function. We present a method using arithmetic nearly-holomorphic forms and general quasi-modular forms, related to algebraic automorphic forms. It gives new technique of constructing -adic zeta-functions via general quasi-modular forms and their Fourier coefficients.
@article{AFST_2016_6_25_2-3_543_0, author = {Panchishkin, Alexei}, title = {Arithmetical modular forms and new constructions of $p$-adic $L$-functions on classical groups}, journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques}, pages = {543--568}, publisher = {Universit\'e Paul Sabatier, Toulouse}, volume = {Ser. 6, 25}, number = {2-3}, year = {2016}, doi = {10.5802/afst.1504}, zbl = {1410.11044}, mrnumber = {3530168}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/afst.1504/} }
TY - JOUR AU - Panchishkin, Alexei TI - Arithmetical modular forms and new constructions of $p$-adic $L$-functions on classical groups JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2016 DA - 2016/// SP - 543 EP - 568 VL - Ser. 6, 25 IS - 2-3 PB - Université Paul Sabatier, Toulouse UR - http://archive.numdam.org/articles/10.5802/afst.1504/ UR - https://zbmath.org/?q=an%3A1410.11044 UR - https://www.ams.org/mathscinet-getitem?mr=3530168 UR - https://doi.org/10.5802/afst.1504 DO - 10.5802/afst.1504 LA - en ID - AFST_2016_6_25_2-3_543_0 ER -
Panchishkin, Alexei. Arithmetical modular forms and new constructions of $p$-adic $L$-functions on classical groups. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 25 (2016) no. 2-3, pp. 543-568. doi : 10.5802/afst.1504. http://archive.numdam.org/articles/10.5802/afst.1504/
[1] Amice ( Y.) and Vélu (J.).— Distributions -adiques associées aux séries de Hecke, Journées Arithmétiques de Bordeaux (Conf. Univ. Bordeaux, 1974), Astérisque no. 24/25, Soc. Math. France, Paris, p. 119-131 (1975). | Numdam | Zbl 0332.14010
[2] Böcherer (S.).— Über die Fourierkoeffizienten Siegelscher Eisensteinreihen, Manuscripta Math., 45, p. 273-288 (1984). | Article | Zbl 0533.10023
[3] Böcherer (S.).— Über die Funktionalgleichung automorpher –Funktionen zur Siegelscher Modulgruppe. J. reine angew. Math. 362, p. 146-168 (1985). | Article | Zbl 0565.10025
[4] Boecherer (S.), Nagaoka (S.).— On -adic properties of Siegel modular forms, arXiv:1305.0604 [math.NT]
[5] Böcherer (S.), Panchishkin (A.A.).— Admissible -adic measures attached to triple products of elliptic cusp forms. Documenta Math. Extra volume : John H.Coates’ Sixtieth Birthday, p. 77-132 (2006). | Zbl 1186.11030
[6] Böcherer (S.), Panchishkin (A.A.).— -adic Interpolation of Triple -functions: Analytic Aspects. In: Automorphic Forms and -functions II: Local Aspects – David Ginzburg, Erez Lapid, and David Soudry, Editors, AMS, BIU, 2009, 313 pp.; p.1-41. | Article
[7] Böcherer (S.), Panchishkin (A.A.).— Higher Twists and Higher Gauss Sums Vietnam Journal of Mathematics 39:3, p. 309-326 (2011).
[8] Böcherer (S.), and Schmidt (C.-G.).— -adic measures attached to Siegel modular forms, Ann. Inst. Fourier 50, No 5, p. 1375-1443 (2000). | Article | MR 1800123 | Zbl 0962.11023
[9] Coates (J.).— On –adic –functions. Sem. Bourbaki, 40eme annee, 1987-88, n 701, Asterisque, p. 177-178 (1989).
[10] Coates (J.) and Perrin-Riou (B.).— On -adic -functions attached to motives over , Advanced Studies in Pure Math. 17, p. 23-54 (1989). | Article
[11] Courtieu (M.), Panchishkin (A.A).— Non-Archimedean -Functions and Arithmetical Siegel Modular Forms, Lecture Notes in Mathematics 1471, Springer-Verlag, 2004 (2nd augmented ed.) | Zbl 1070.11023
[12] Eischen (E. E.).— -adic Differential Operators on Automorphic Forms on Unitary Groups. Annales de l’Institut Fourier 62. No 1, p. 177-243 (2012). | Article | MR 2986270 | Zbl 1257.11054
[13] Eischen (E. E.), Harris (M.), Li (J-S), Skinner (C.M.).— -adic -functions for Unitary Shimura Varieties, II , part II: zeta-integral calculations. (Submitted on 4 Feb 2016), 73 pages, arXiv:1602.01776 [math.NT]
[14] Gelbart (S.), and Shahidi (F.).— Analytic Properties of Automorphic -functions, Academic Press, New York, (1988). | Zbl 0654.10028
[15] Gelbart (S.), Piatetski-Shapiro (I.I.), Rallis (S.).— Explicit constructions of automorphic - functions. Springer-Verlag, Lect. Notes in Math. N 1254 152p. (1987). | Article | Zbl 0612.10022
[16] Guerzhoy (P.).— On -adic families of Siegel cusp forms in the Maass Spezialschaar. Journal für die reine und angewandte Mathematik 523, p. 103-112 (2000). | Article | Zbl 0944.11015
[17] Harris (M.).— The rationality of holomorphic Eisenstein series, Inv. Math. 63, p. 305-310 (1981). | Article | MR 610541 | Zbl 0452.10031
[18] Harris (M.).— Eisenstein Series on Shimura Varieties. Ann. Math., 119, No. 1 p. 59-94 (1984). | Article | MR 736560
[19] Harris (M.), Li (-S), Skinner (Ch.M.).— -adic -functions for unitary Shimura varieties. Documenta Math. Extra volume : John H.Coates’ Sixtieth Birthday, p. 393-464 (2006).
[20] Hecke (E.).— Theorie der Eisensteinschen Reihen und ihre Anwebdung auf Funktionnentheorie und Arithmetik, Abh. Math. Sem. Hamburg 5), p. 199-224 (1927. | MR 3069476 | Zbl 53.0345.02
[21] Hida (H.).— Elementary theory of -functions and Eisenstein series. London Mathematical Society Student Texts. 26 Cambridge (1993). | Article | Zbl 0942.11024
[22] Hida (H.).— Control theorems for coherent sheaves on Shimura varieties of PEL-type, Journal of the Inst. of Math. Jussieu 1, p. 1-76 (2002). | Article | MR 1954939 | Zbl 1039.11041
[23] Ichikawa (T.).— Vector-valued -adic Siegel modular forms, J. reine angew. Math., DOI 10.1515/ crelle-2012-0066. | Article | MR 3200333 | Zbl 1361.11031
[24] Ichikawa (T.).— Arithmeticity of vector-valued Siegel modular forms in analytic and p-adic cases. Preprint, 2013
[25] Ikeda (T.).— On the lifting of elliptic cusp forms to Siegel cusp forms of degree , Ann. of Math. (2) 154, p. 641-681 (2001). | Article | MR 1884618 | Zbl 0998.11023
[26] Ikeda (T.).— , Pullback of the lifting of elliptic cusp forms and Miyawaki’s Conjecture Duke Mathematical Journal, 131, p. 469-497 (2006). | Article | MR 2219248 | Zbl 1112.11022
[27] Katz (N.M.).— -adic interpolation of real analytic Eisenstein series. Ann. of Math. 104, p. 459-571 (1976). | Article | MR 506271 | Zbl 0354.14007
[28] Kawamura (H.-A.).— On certain constructions of -adic families of Siegel modular forms of even genus ArXiv, 1011.6042v1
[29] Kazhdan (D.), Mazur (B.), Schmidt (C.-G.).— Relative modular symbols and Rankin-Selberg convolutions. J. Reine Angew. Math. 519, p. 97-141 (2000). | Article | MR 1739728 | Zbl 0941.11020
[30] Klingen (H.).— Über die Werte der Dedekindschen Zetafunktionen. Math. Ann. 145, p. 265-272 (1962). | Article | Zbl 0101.03002
[31] Klingen (H.).— Zum Darstellungssatz für Siegelsche Modulformen. Math. Z. 102, p. 30-43 (1967). | Article | Zbl 0155.40401
[32] Kubota (T.).— Elementary Theory of Eisenstein Series, Kodansha Ltd. and John Wiley and Sons (Halsted Press), (1973).
[33] Lang (S.).— Introduction to modular forms. With appendixes by D. Zagier and Walter Feit. Springer-Verlag, Berlin, (1995).
[34] Maass (H.).— Siegel’s modular forms and Dirichlet series Springer-Verlag, Lect. Notes in Math. N 216 (1971). | Article | Zbl 0224.10028
[35] Miyake, Toshitsune.— Modular forms. Transl. from the Japanese by Yoshitaka Maeda., Berlin etc.: Springer-Verlag. viii, 335 p. (1989). | Article | Zbl 0701.11014
[36] Miyawaki (I.).— Numerical examples of Siegel cusp forms of degree 3 and their zeta-functions, Memoirs of the Faculty of Science, Kyushu University, Ser. A, Vol. 46, No. 2, p. 307-339 (1992). | Article | MR 1195472 | Zbl 0780.11022
[37] Mumford (D.).— An analytic construction of degenerating abelian varieties over complete rings, Compositio Math. 24, p. 239-272 (1972). | MR 352106 | Zbl 0241.14020
[38] Panchishkin (A.A.).— Complex valued measures attached to Euler products, Trudy Sem. Petrovskogo 7 (1981) p. 239-244 (in Russian) | Zbl 0496.10016
[39] Pančiškin (A.A.).— Le prolongement -adique analytique de fonctions de Rankin I,II. C. R. Acad. Sci. Paris 294, 51-53, p. 227-230 (1982).
[40] Panchishkin (A.A.).— Non–Archimedean -functions of Siegel and Hilbert modular forms, Lecture Notes in Math., 1471, Springer–Verlag, (1991), 166p. | Article | MR 1122593 | Zbl 0732.11026
[41] Panchishkin (A.A.).— Admissible Non-Archimedean standard zeta functions of Siegel modular forms, Proceedings of the Joint AMS Summer Conference on Motives, Seattle, July 20-August 2 1991, Seattle, Providence, R.I., 1994, vol.2, p. 251-292. | Article | Zbl 0837.11029
[42] Panchishkin (A.A.).— On the Siegel-Eisenstein measure and its applications, Israel Journal of Mathemetics, 120, Part B, p. 467-509 (2000). | Article | MR 1809631 | Zbl 0977.11021
[43] Panchishkin (A.A.).— A new method of constructing -adic -functions associated with modular forms, Moscow Mathematical Journal, 2, Number 2, p. 1-16 (2002). | Article | Zbl 1011.11026
[44] Panchishkin (A.A.).— Two variable -adic functions attached to eigenfamilies of positive slope, Invent. Math. v. 154, N3, p. 551-615 (2003). | Article | MR 2018785 | Zbl 1065.11025
[45] Panchishkin (A.A.).— On -adic integration in spaces of modular forms and its applications, J. Math. Sci., New York 115, No.3, p. 2357-2377 (2003). | Article | MR 1981306 | Zbl 1040.11034
[46] Panchishkin (A.A.).— Two modularity lifting conjectures for families of Siegel modular forms, Mathematical Notes Volume 88 Numbers 3-4, p. 544-551 (2010). | Article | Zbl 1245.11062
[47] Panchishkin (A.A.).— Families of Siegel modular forms, -functions and modularity lifting conjectures. Israel Journal of Mathemetics, 185, p. 343-368 (2011). | Article | MR 2837140 | Zbl 1277.11047
[48] Serre (J.-P.).— Formes modulaires et fonctions zêta -adiques, Lect Notes in Math. 350 p. 191-268 (Springer Verlag) (1973). | Article
[49] Siegel (C. L)..— Über die analytische Theorie der quadratischen Formen. Ann. of Math. 36, p. 527-606 (1935). | Article | Zbl 61.0140.01
[50] Siegel (C. L.).— Einführung in die Theorie der Modulfunktionen -ten Grades. Math. Ann. 116, p. 617-657 (1939). | Article | Zbl 65.0357.01
[51] Shimura (G.).— Eisenstein series and zeta functions on symplectic groups, Inventiones Math. 119, p. 539-584 (1995). | Article | MR 1317650 | Zbl 0845.11020
[52] Shimura (G.).— Arithmeticity in the theory of automorphic forms, Mathematical Surveys and Monographs, vol. 82 (Amer. Math. Soc., Providence, 2000). | Zbl 0967.11001
[53] Skinner (C.) and Urban (E.) .— The Iwasawa Main Cconjecture for GL(2). http://www.math.jussieu.fr/~urban/eurp/MC.pdf
[54] Sturm (J.).— The critical values of zeta-functions associated to the symplectic group. Duke Math. J. 48, p. 327-350 (1981). | Article | MR 620253 | Zbl 0483.10026
[55] Urban (E.).— Nearly Overconvergent Modular Forms, Iwasawa Theory 2012. Contributions in Mathematical and Computational Sciences Volume 7, 2014, p. 401-441, Date: 12 Nov 2014 http://link.springer.com/chapter/10.1007/978-3-642-55245-8_14 | Article | MR 3586822 | Zbl 1328.11052
[56] Washington (L.).— Introduction to cyclotomic fields, Springer Verlag: N.Y. e.a., 1982 | Article | Zbl 0484.12001
[57] Yifan (H.).— The 4th largest tree in the Mathematical Genealogy Project, Produced by Yifan Hu, AT&T Shannon Laboratory. http://yifanhu.net/GALLERY/MATHGENEALOGY/plot_comp4_p2.2_font8.pdf
[58] Yoshida (H.).— Review on Goro Shimura, Arithmeticity in the theory of automorphic forms [52], Bulletin (New Series) of the AMS, vol. 39, N3), p. 441-448 (2002). | Article
Cité par Sources :