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Rankin’s Lemma of Higher Genus and Explicit Formulas for Hecke Operators

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Book cover Algebra, Arithmetic, and Geometry

Part of the book series: Progress in Mathematics ((PM,volume 270))

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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 GSp2 ×GSp2 to GSp4 is given using Ikeda–Miyawaki constructions.

2000 Mathematics Subject Classifications: 11F46, 11F66

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References

  1. Andrianov, A.N., Shimura’s conjecture for Siegel’s modular group of genus 3, Dokl. Akad. Nauk SSSR 177 (1967), 755–758 = Soviet Math. Dokl. 8 (1967), 1474–1478.

    MathSciNet  Google Scholar 

  2. Andrianov, A.N., 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 (1968), 9–11 = Soviet Math. Dokl. 9 (1968), 1295–1297.

    MathSciNet  Google Scholar 

  3. Andrianov, A.N., 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 (1969), No. 3 (Math. USSR – Izvestija, Vol. 3 (1969), No. 3, pp. 439–476).

    MathSciNet  Google Scholar 

  4. Andrianov, A.N., Spherical functions for GL n over local fields and summation of Hecke series, Mat. Sbornik, Tom 83 (125) (1970), No 3 (Math. USSR Sbornik, Vol. 12 (1970), No. 3, pp. 429–452).

    Article  MathSciNet  Google Scholar 

  5. Andrianov, A.N., Euler products corresponding to Siegel modular forms of genus 2, Russian Math. Surveys, 29:3 (1974), pp. 45–116 (Uspekhi Mat. Nauk 29:3 (1974) pp. 43–110).

    Article  Google Scholar 

  6. Andrianov, A.N., Quadratic Forms and Hecke Operators, Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1987.

    MATH  Google Scholar 

  7. Andrianov, A.N., Kalinin, V.L., On Analytic Properties of Standard Zeta Functions of Siegel Modular Forms, Mat. Sbornik 106 (1978), pp 323–339 (in Russian).

    MathSciNet  Google Scholar 

  8. Andrianov, A.N., Zhuravlev, V.G., Modular Forms and Hecke Operators, Translations of Mathematical Monographs, Vol. 145, AMS, Providence, Rhode Island, 1995.

    Google Scholar 

  9. Asgari, M., Schmidt, R., Siegel modular forms and representations. Manuscripta Math. 104 (2001), 173–200.

    Article  MATH  MathSciNet  Google Scholar 

  10. Böcherer, S., Ein Rationalitätssatz für formale Heckereihen zur Siegelschen Modulgruppe. Abh. Math. Sem. Univ. Hamburg 56 (1986), 35–47.

    Article  MATH  MathSciNet  Google Scholar 

  11. Böcherer, S. and Heim B.E. Critical values of L-functions on GSp 2 × GL 2. Math.Z, 254, 485–503 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  12. Bump, D.; Friedberg, S.; Ginzburg, D. Lifting automorphic representations on the double covers of orthogonal groups. Duke Math. J. 131 (2006), no. 2, 363–396.

    Article  MATH  MathSciNet  Google Scholar 

  13. Bump, D.; Friedberg, S.; Ginzburg, D. Whittaker-orthogonal models, functoriality, and the Rankin-Selberg method. Invent. Math. 109 (1992), no. 1, 55–96.

    Article  MATH  MathSciNet  Google Scholar 

  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.).

    Google Scholar 

  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

    Google Scholar 

  16. Evdokimov, S.A., Dirichlet series, multiple Andrianov zeta-functions in the theory of Euler modular forms of genus 3 (Russian), Dokl. Akad. Nauk SSSR 277 (1984), no. 1, 25–29.

    MathSciNet  Google Scholar 

  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.

    Google Scholar 

  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).

    Google Scholar 

  19. Ikeda, T., On the lifting of elliptic cusp forms to Siegel cusp forms of degree 2n, Ann. of Math. (2) 154 (2001), 641–681.

    Article  MATH  MathSciNet  Google Scholar 

  20. Ikeda, T., Pullback of the lifting of elliptic cusp forms and Miyawaki’s Conjecture Duke Mathematical Journal, 131, 469–497 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  21. Jiang, D., Degree 16 standard L-function of GSp (2) × GSp (2). Mem. Amer. Math. Soc. 123 (1996), no. 588.

    Google Scholar 

  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.

    Google Scholar 

  23. R. Langlands, Automorphic forms, Shimura varieties, and L-functions. Ein Märchen. Proc. Symp. Pure Math. 33, part 2 (1979), 205–246.

    Google Scholar 

  24. Maass, H., Indefinite Quadratische Formen und Eulerprodukte. Comm. on Pure and Appl. Math, 19, 689–699 (1976).

    Article  MathSciNet  Google Scholar 

  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.

    Google Scholar 

  26. Manin, Yu.I. and Panchishkin, A.A., Introduction to Modern Number Theory, Encyclopaedia of Mathematical Sciences, vol. 49 (2nd ed.), Springer-Verlag, 2005.

    Google Scholar 

  27. Murokawa, K., Relations between symmetric power L-functions and spinor L-functions attached to Ikeda lifts, Kodai Math. J. 25, 61–71 (2002).

    Article  MathSciNet  Google Scholar 

  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.

    Google Scholar 

  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.

    Google Scholar 

  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.

    Google Scholar 

  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.

    Google Scholar 

  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.

    Google Scholar 

  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.

    Google Scholar 

  34. Panchishkin, A., Vankov, K. Explicit Shimura’s conjecture for Sp3 on a computer. Math. Res. Lett. 14, No. 2, (2007), 173–187.

    Google Scholar 

  35. Schmidt, R., On the spin L-function of Ikeda’s lifts. Comment. Math. Univ. St. Pauli 52 (2003), 1–46.

    MATH  MathSciNet  Google Scholar 

  36. Satake, I., Theory of spherical functions on reductive groups over p-adic fields, Publ. mathématiques de l’IHES 18 (1963), 1–59.

    Google Scholar 

  37. Shimura, G., On modular correspondences for Sp(n,Z) and their congruence relations, Proc. Nat. Acad. Sci. U.S.A. 49 (1963), 824–828.

    Article  MATH  MathSciNet  Google Scholar 

  38. Shimura, G., Introduction to the Arithmetic Theory of Automorphic Functions, Princeton Univ. Press, 1971.

    Google Scholar 

  39. Tamagawa, T., On the ζ-function of a division algebra, Ann. of Math. 77 (1963), 387–405.

    Article  MathSciNet  Google Scholar 

  40. Vankov, K., Explicit formula for the symplectic Hecke series of genus four, Arxiv, math.NT/0606492 (2006).

    Google Scholar 

  41. Yoshida, H., Siegel’s Modular Forms and the Arithmetic of Quadratic Forms, Inventiones math. 60, 193–248 (1980).

    Article  MATH  Google Scholar 

  42. Yoshida, H., Motives and Siegel modular forms, American Journal of Mathematics, 123 (2001), 1171–1197.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institut Fourier, Université de Grenoble I, BP 74, 38402, Saint-Martin d’Hères, France

    Alexei Panchishkin & Kirill Vankov

Authors
  1. Alexei Panchishkin
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  2. Kirill Vankov
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Correspondence to Alexei Panchishkin .

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Editors and Affiliations

  1. Courant Inst. Mathematical, New York University, Mercer St. 251, New York, 10012, USA

    Yuri Tschinkel

  2. Eberly College of Science, Pennsylvania State University, University Park, 16802, USA

    Yuri Zarhin

Additional information

To dear Yuri Ivanovich Manin for his seventieth birthday with admiration

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Panchishkin, A., Vankov, K. (2009). Rankin’s Lemma of Higher Genus and Explicit Formulas for Hecke Operators. In: Tschinkel, Y., Zarhin, Y. (eds) Algebra, Arithmetic, and Geometry. Progress in Mathematics, vol 270. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4747-6_17

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