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Rankin’s lemma of higher genus and explicit formulas for Hecke operators. (English) Zbl 1246.11105

Tschinkel, Yuri (ed.) et al., Algebra, arithmetic, and geometry. In honor of Y. I. Manin on the occasion of his 70th birthday. Vol. II. Boston, MA: Birkhäuser (ISBN 978-0-8176-4746-9/hbk; 978-0-8176-4747-6/ebook). Progress in Mathematics 270, 533-554 (2009).
Summary: We develop explicit formulas for Hecke operators of higher genus in terms of spherical coordinates. Applications are given to summation of various generating series with coefficients in local Hecke algebras and in a tensor product of such algebras. In particular, we formulate and prove Rankin’s lemma in genus two. An application to a holomorphic lifting from \(\text{GSp}_2\times\text{GSp}_2\) to \(\text{GSp}_4\) is given using Ikeda-Miyawaki constructions.
For the entire collection see [Zbl 1185.00042].

MSC:

11F60 Hecke-Petersson operators, differential operators (several variables)
11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
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[1] Andrianov, AN, Shimura’s conjecture for Siegel’s modular group of genus 3, Dokl. Akad. Nauk SSSR, 177, 755-758 (1967) · Zbl 0167.07001
[2] Andrianov, AN, Rationality of multiple Hecke series of a complete linear group and Shimura’s hypothesis on Hecke’s series of a symplectic group, Dokl. Akad. Nauk SSSR, 183, 9-11 (1968) · Zbl 0191.51703
[3] Andrianov, AN, Rationality theorems for Hecke series and zeta functions of the groups GL_n and SP_n over local fields, Izv. Akad. Nauk SSSR, Ser. Mat., Tom, 33, 439-476 (1969) · Zbl 0236.20033
[4] Andrianov, AN, Spherical functions for GL_n over local fields and summation of Hecke series, Mat. Sbornik, Tom, 83, 125, 429-452 (1970) · Zbl 0272.22008 · doi:10.1070/SM1970v012n03ABEH000929
[5] Andrianov, AN, Euler products corresponding to Siegel modular forms of genus 2, Russian Math. Surveys, 29, 3 (1974) · Zbl 0304.10021 · doi:10.1070/RM1974v029n03ABEH001285
[6] Andrianov, AN, Quadratic Forms and Hecke Operators (1987), Berlin, Heidelberg, New York, London, Paris, Tokyo: Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo · Zbl 0613.10023
[7] Andrianov, AN; Kalinin, VL, On Analytic Properties of Standard Zeta Functions of Siegel Modular Forms, Mat. Sbornik, 106, 323-339 (1978) · Zbl 0389.10023
[8] Andrianov, A.N., Zhuravlev, V.G., Modular Forms and Hecke Operators, Translations of Mathematical Monographs, Vol. 145, AMS, Providence, Rhode Island, 1995. · Zbl 0838.11032
[9] Asgari, M.; Schmidt, R., Siegel modular forms and representations, Manuscripta Math, 104, 173-200 (2001) · Zbl 0987.11037 · doi:10.1007/PL00005869
[10] Böcherer, S., Ein Rationalitätssatz für formale Heckereihen zur Siegelschen Modulgruppe, Abh. Math. Sem. Univ. Hamburg, 56, 35-47 (1986) · Zbl 0613.10026 · doi:10.1007/BF02941505
[11] Böcherer, S.; Heim, BE, Critical values of L-functions on GSp_2 × GL_2, Math.Z, 254, 485-503 (2006) · Zbl 1197.11059 · doi:10.1007/s00209-005-0945-z
[12] Bump, D.; Friedberg, S.; Ginzburg, D., Lifting automorphic representations on the double covers of orthogonal groups, Duke Math. J., 131, 363-396 (2006) · Zbl 1107.11024 · doi:10.1215/S0012-7094-06-13126-5
[13] Bump, D.; Friedberg, S.; Ginzburg, D., Whittaker-orthogonal models, functoriality, and the Rankin-Selberg method, Invent. Math., 109, 55-96 (1992) · Zbl 0782.11015 · doi:10.1007/BF01232019
[14] 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.). · Zbl 1070.11023
[15] Deligne P., Valeurs de fonctions L et périodes d’intégrales, Proc. Sympos. Pure Math. vol. 55. Amer. Math. Soc., Providence, RI, 1979, 313-346 · Zbl 0449.10022
[16] Evdokimov, SA, Dirichlet series, multiple Andrianov zeta-functions in the theory of Euler modular forms of genus 3, (Russian), Dokl. Akad. Nauk SSSR, 277, 25-29 (1984)
[17] Faber, C., van der Geer, G., Sur la cohomologie des systmes locaux sur les espaces de modules des courbes de genre 2 et des surfaces abliennes. I, II C. R. Math. Acad. Sci. Paris 338, (2004) No.5, pp. 381-384 and No.6, 467-470. · Zbl 1055.14026
[18] Hecke, E., Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktenwickelung, I, II. Math. Annalen 114 (1937), 1-28, 316-351 (Mathematische Werke. Göttingen: Vandenhoeck und Ruprecht, 1959, 644-707). · Zbl 0016.35503
[19] Ikeda, T., On the lifting of elliptic cusp forms to Siegel cusp forms of degree 2n, Ann. of Math. (2), 154, 641-681 (2001) · Zbl 0998.11023 · doi:10.2307/3062143
[20] Ikeda, T., Pullback of the lifting of elliptic cusp forms and Miyawaki’s Conjecture, Duke Mathematical Journal, 131, 469-497 (2006) · Zbl 1112.11022 · doi:10.1215/S0012-7094-06-13133-2
[21] Jiang, D., Degree 16 standard L-function of GSp (2) × GSp (2). Mem. Amer. Math. Soc. 123 (1996), no. 588. · Zbl 0874.11040
[22] Kurokawa, N., Analyticity of Dirichlet series over prime powers. Analytic number theory (Tokyo, 1988), 168-177, Lecture Notes in Math., 1434, Springer, Berlin, 1990. · Zbl 0707.11063
[23] R. Langlands, Automorphic forms, Shimura varieties, and L-functions. Ein Märchen. Proc. Symp. Pure Math. 33, part 2 (1979), 205-246. · Zbl 0447.12009
[24] Maass, H., Indefinite Quadratische Formen und Eulerprodukte, Comm. on Pure and Appl. Math, 19, 689-699 (1976) · Zbl 0326.10021 · doi:10.1002/cpa.3160290610
[25] Manin, Yu.I. and Panchishkin, A.A., Convolutions of Hecke series and their values at integral points, Mat. Sbornik, 104 (1977) 617-651 (in Russian); Math. USSR, Sb. 33, 539-571 (1977) (English translation). In: Selected Papers of Yu. I.Manin, World Scientific, 1996, pp. 325-357.
[26] Manin, Yu.I. and Panchishkin, A.A., Introduction to Modern Number Theory, Encyclopaedia of Mathematical Sciences, vol. 49 (2nd ed.), Springer-Verlag, 2005. · Zbl 1079.11002
[27] Murokawa, K., Relations between symmetric power L-functions and spinor L-functions attached to Ikeda lifts, Kodai Math. J., 25, 61-71 (2002) · Zbl 1015.11016 · doi:10.2996/kmj/1106171076
[28] Panchishkin, A., A functional equation of the non-Archimedean Rankin convolution, Duke Math. J. 54 (special volume in Honor of Yu.I. Manin on his 50th birthday) (1987) 77-89. · Zbl 0633.10028
[29] Panchishkin, 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, 251-292. · Zbl 0837.11029
[30] Panchishkin, A., A new method of constructing p-adic L-functions associated with modular forms, Moscow Mathematical Journal, 2 (2002), Number 2 (special issue in Honor of Yu.I. Manin on his 65th birthday), 1-16. · Zbl 1011.11026
[31] Panchishkin, A., Produits d’Euler attachés aux formes modulaires de Siegel. Exposé au Séminaire Groupes Réductifs et Formes Automorphes à l’Institut de Mathématiques de Jussieu, June 22, 2006.
[32] Panchishkin, A.A., Triple products of Coleman’s families and their periods (a joint work with S. Boecherer) Proceedings of the 8th Hakuba conference “Periods and related topics from automorphic forms,” September 25-October 1, 2005.
[33] Panchishkin, A.A., p-adic Banach modules of arithmetical modular forms and triple products of Coleman’s families, (for a special volume of Quarterly Journal of Pure and Applied Mathematics dedicated to Jean-Pierre Serre), 2006.
[34] Panchishkin, A., Vankov, K. Explicit Shimura’s conjecture for Sp3 on a computer. Math. Res. Lett. 14, No. 2, (2007), 173-187. · Zbl 1163.11039
[35] Schmidt, R., On the spin L-function of Ikeda’s lifts, Comment. Math. Univ. St. Pauli, 52, 1-46 (2003) · Zbl 1195.11064
[36] Satake, I., Theory of spherical functions on reductive groups over p-adic fields, Publ. mathématiques de l’IHES, 18, 1-59 (1963)
[37] Shimura, G., On modular correspondences for Sp(n,Z) and their congruence relations, Proc. Nat. Acad. Sci. U.S.A., 49, 824-828 (1963) · Zbl 0122.08803 · doi:10.1073/pnas.49.6.824
[38] Shimura, G., Introduction to the Arithmetic Theory of Automorphic Functions, Princeton Univ. Press, 1971. · Zbl 0221.10029
[39] Tamagawa, T., On the ζ-function of a division algebra, Ann. of Math., 77, 387-405 (1963) · Zbl 0222.12018 · doi:10.2307/1970221
[40] Vankov, K., Explicit formula for the symplectic Hecke series of genus four, Arxiv, math.NT/0606492 (2006).
[41] Yoshida, H., Siegel’s Modular Forms and the Arithmetic of Quadratic Forms, Inventiones math., 60, 193-248 (1980) · Zbl 0453.10022 · doi:10.1007/BF01390016
[42] Yoshida, H., Motives and Siegel modular forms, American Journal of Mathematics, 123, 1171-1197 (2001) · Zbl 0998.11022 · doi:10.1353/ajm.2001.0042
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