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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.

Une nouvelle approche pour construire des fonctions L p-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 G sur un corps de nombres K les fonctions L complexes sont certains produits d’Euler L(s,π,r,χ). En particulier, notre construction couvre les fonctions L étudiées par Shimura dans [52] via la méthode de doublement de Piatetski-Shapiro et Rallis. Un avatar p-adique L(s,π,r,χ) est une fonction p-adique analytique L p (s,π,r,χ) de s p , χmodp r interpolant les valeurs spéciales normalisées algébriques L * (s,π,r,χ) de la fonction L 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 L-functions and their p-adic avatars is presented as a work in progress with Thanh Hung Dang and Anh Tuan Do (Hanoi, Vietnam). For an algebraic group G over a number field K these L functions are certain Euler products L(s,π,r,χ). In particular, our constructions cover the L-functions in [52] via the doubling method of Piatetski-Shapiro and Rallis.

A p-adic avatar of L(s,π,r,χ) is a p-adic analytic function L p (s,π,r,χ) of p-adic arguments s p , χmodp r which interpolates algebraic numbers defined through the normalized critical values L * (s,π,r,χ) of the corresponding complex analytic L-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 p-adic zeta-functions via general quasi-modular forms and their Fourier coefficients.

Publié le : 2016-07-11
DOI : https://doi.org/10.5802/afst.1504
@article{AFST_2016_6_25_2-3_543_0,
     author = {Alexei Panchishkin},
     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},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 25},
     number = {2-3},
     year = {2016},
     pages = {543-568},
     doi = {10.5802/afst.1504},
     language = {en},
     url = {afst.centre-mersenne.org/item/AFST_2016_6_25_2-3_543_0/}
}
Alexei Panchishkin. 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. https://afst.centre-mersenne.org/item/AFST_2016_6_25_2-3_543_0/

[1] Amice ( Y.) and Vélu (J.).— Distributions p-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).

[2] Böcherer (S.).— Über die Fourierkoeffizienten Siegelscher Eisensteinreihen, Manuscripta Math., 45, p. 273-288 (1984).

[3] Böcherer (S.).— Über die Funktionalgleichung automorpher L–Funktionen zur Siegelscher Modulgruppe. J. reine angew. Math. 362, p. 146-168 (1985).

[4] Boecherer (S.), Nagaoka (S.).— On p-adic properties of Siegel modular forms, arXiv:1305.0604 [math.NT]

[5] Böcherer (S.), Panchishkin (A.A.).— Admissible p-adic measures attached to triple products of elliptic cusp forms. Documenta Math. Extra volume : John H.Coates’ Sixtieth Birthday, p. 77-132 (2006).

[6] Böcherer (S.), Panchishkin (A.A.).— p-adic Interpolation of Triple L-functions: Analytic Aspects. In: Automorphic Forms and L-functions II: Local Aspects – David Ginzburg, Erez Lapid, and David Soudry, Editors, AMS, BIU, 2009, 313 pp.; p.1-41.

[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.).— p-adic measures attached to Siegel modular forms, Ann. Inst. Fourier 50, No 5, p. 1375-1443 (2000).

[9] Coates (J.).— On p–adic L–functions. Sem. Bourbaki, 40eme annee, 1987-88, n 701, Asterisque, p. 177-178 (1989).

[10] Coates (J.) and Perrin-Riou (B.).— On p-adic L-functions attached to motives over , Advanced Studies in Pure Math. 17, p. 23-54 (1989).

[11] 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.)

[12] Eischen (E. E.).— p-adic Differential Operators on Automorphic Forms on Unitary Groups. Annales de l’Institut Fourier 62. No 1, p. 177-243 (2012).

[13] Eischen (E. E.), Harris (M.), Li (J-S), Skinner (C.M.).— p-adic L-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 L-functions, Academic Press, New York, (1988).

[15] Gelbart (S.), Piatetski-Shapiro (I.I.), Rallis (S.).— Explicit constructions of automorphic L - functions. Springer-Verlag, Lect. Notes in Math. N 1254 152p. (1987).

[16] Guerzhoy (P.).— On p-adic families of Siegel cusp forms in the Maass Spezialschaar. Journal für die reine und angewandte Mathematik 523, p. 103-112 (2000).

[17] Harris (M.).— The rationality of holomorphic Eisenstein series, Inv. Math. 63, p. 305-310 (1981).

[18] Harris (M.).— Eisenstein Series on Shimura Varieties. Ann. Math., 119, No. 1 p. 59-94 (1984).

[19] Harris (M.), Li (-S), Skinner (Ch.M.).— p-adic L-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.

[21] Hida (H.).— Elementary theory of L-functions and Eisenstein series. London Mathematical Society Student Texts. 26 Cambridge (1993).

[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).

[23] Ichikawa (T.).— Vector-valued p-adic Siegel modular forms, J. reine angew. Math., DOI 10.1515/ crelle-2012-0066.

[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 2n, Ann. of Math. (2) 154, p. 641-681 (2001).

[26] Ikeda (T.).— , Pullback of the lifting of elliptic cusp forms and Miyawaki’s Conjecture Duke Mathematical Journal, 131, p. 469-497 (2006).

[27] Katz (N.M.).— p-adic interpolation of real analytic Eisenstein series. Ann. of Math. 104, p. 459-571 (1976).

[28] Kawamura (H.-A.).— On certain constructions of p-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).

[30] Klingen (H.).— Über die Werte der Dedekindschen Zetafunktionen. Math. Ann. 145, p. 265-272 (1962).

[31] Klingen (H.).— Zum Darstellungssatz für Siegelsche Modulformen. Math. Z. 102, p. 30-43 (1967).

[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).

[35] Miyake, Toshitsune.— Modular forms. Transl. from the Japanese by Yoshitaka Maeda., Berlin etc.: Springer-Verlag. viii, 335 p. (1989).

[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).

[37] Mumford (D.).— An analytic construction of degenerating abelian varieties over complete rings, Compositio Math. 24, p. 239-272 (1972).

[38] Panchishkin (A.A.).— Complex valued measures attached to Euler products, Trudy Sem. Petrovskogo 7 (1981) p. 239-244 (in Russian)

[39] Pančiškin (A.A.).— Le prolongement p-adique analytique de fonctions L de Rankin I,II. C. R. Acad. Sci. Paris 294, 51-53, p. 227-230 (1982).

[40] Panchishkin (A.A.).— Non–Archimedean L-functions of Siegel and Hilbert modular forms, Lecture Notes in Math., 1471, Springer–Verlag, (1991), 166p.

[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.

[42] Panchishkin (A.A.).— On the Siegel-Eisenstein measure and its applications, Israel Journal of Mathemetics, 120, Part B, p. 467-509 (2000).

[43] Panchishkin (A.A.).— A new method of constructing p-adic L-functions associated with modular forms, Moscow Mathematical Journal, 2, Number 2, p. 1-16 (2002).

[44] Panchishkin (A.A.).— Two variable p-adic L functions attached to eigenfamilies of positive slope, Invent. Math. v. 154, N3, p. 551-615 (2003).

[45] Panchishkin (A.A.).— On p-adic integration in spaces of modular forms and its applications, J. Math. Sci., New York 115, No.3, p. 2357-2377 (2003).

[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).

[47] Panchishkin (A.A.).— Families of Siegel modular forms, L-functions and modularity lifting conjectures. Israel Journal of Mathemetics, 185, p. 343-368 (2011).

[48] Serre (J.-P.).— Formes modulaires et fonctions zêta p-adiques, Lect Notes in Math. 350 p. 191-268 (Springer Verlag) (1973).

[49] Siegel (C. L)..— Über die analytische Theorie der quadratischen Formen. Ann. of Math. 36, p. 527-606 (1935).

[50] Siegel (C. L.).— Einführung in die Theorie der Modulfunktionen n-ten Grades. Math. Ann. 116, p. 617-657 (1939).

[51] Shimura (G.).— Eisenstein series and zeta functions on symplectic groups, Inventiones Math. 119, p. 539-584 (1995).

[52] Shimura (G.).— Arithmeticity in the theory of automorphic forms, Mathematical Surveys and Monographs, vol. 82 (Amer. Math. Soc., Providence, 2000).

[53] Skinner (C.) and Urban (E.) .— The Iwasawa Main Cconjecture for GL(2).

[54] Sturm (J.).— The critical values of zeta-functions associated to the symplectic group. Duke Math. J. 48, p. 327-350 (1981).

[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

[56] Washington (L.).— Introduction to cyclotomic fields, Springer Verlag: N.Y. e.a., 1982

[57] Yifan (H.).— The 4th largest tree in the Mathematical Genealogy Project, Produced by Yifan Hu, AT&T Shannon Laboratory.

[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).