L-Functions for GSp(2) × GL(2): Archimedean Theory and Applications

Moriyama, Tomonori

Canadian journal of mathematics, Tome 61 (2009) no. 2, pp. 395-426

Résumé

Let $\Pi$ be a generic cuspidal automorphic representation of $\text{GSp}\left( 2 \right)$ defined over a totally real algebraic number field $\text{k}$ whose archimedean type is either a (limit of) large discrete series representation or a certain principal series representation. Through explicit computation of archimedean local zeta integrals, we prove the functional equation of tensor product $L$ -functions $L\left( s,\Pi \times \sigma\right)$ for an arbitrary cuspidal automorphic representation $\sigma $ of $\text{GL}\left( 2 \right)$ . We also give an application to the spinor $L$ -function of $\Pi$ .

Détail

DOI : 10.4153/CJM-2009-021-x

Mots-clés : 11F70, 11F41, 11F46

Bibliographie

[A-S-1] [A-S-1] Asgari, M. and Shahidi, F., Generic transfer for general spin groups. Duke Math. J. 132(2006), 137–190. Google Scholar

[A-S-2] [A-S-2] Asgari, M. and Shahidi, F., Generic transfer from GSp(4) to GL(4. Compos. Math. 142(2006), 541–550. Google Scholar

[A-G-R] [A-G-R] Ash, A., Ginzburg, D., and Rallis, S., Vanishing periods of cusp forms over modular symbols , Math. Ann. 296(1993), no. 4, 709–723. Google Scholar

[Ba-M] [Ba-M] Barbasch, D. and Moy, A., Whittaker models with an Iwahori fixed vector. In: Representation theory and analysis on homogeneous spaces, Contemp. Math. 177, American Mathematical Soiety, Providence, RI, 1994, pp. 101–105. Google Scholar

[Bo-He] [Bo-He] Böcherer, S. and Heim, B., L-functions on GSp2 × Gl2 of mixed weights. Math. Z. 235(2000), no. 1, 11–51. Google Scholar

[Bo] [Bo] Borel, A., Automorphic L-functions. In: Automorphic forms, representations and L -functions, Proc. Sympos. Pure Math. 33, American Mathematical Society, Providence, RI, 1979, pp. 27–61. Google Scholar

[Bu] [Bu] Bump, D., The Rankin-Selberg method: a survey. In: Number theory, trace formulas and discrete groups, Academic Press, Boston, MA, 1989, pp. 49–109. Google Scholar

[C-S] [C-S] Casselman, W. and Shalika, J., The unramified principal series of p-adic groups. II. The Whittaker function. Compositio Math. 41(1980), no. 2, 207–231. Google Scholar

[Co] [Co] Cogdell, J. W., Lectures on L-functions, converse theorems, and functoriality for GLn. In: Lectures on automorphic L -functions, Fields Inst. Monogr.20, American Mathematical Society, Providence, RI, 2004, pp. 1–96. Google Scholar

[Fu] [Fu] Furusawa, M., On L-functions for GSp(4) × GL(2) and their special values. J. Reine Angew. Math. 438(1993), 187–218. Google Scholar

[G-PS-R] [G-PS-R] Gelbart, S., Piatetski-Shapiro, I., and Rallis, S., Explicit Constructions of Automorphic L-functions. Lecture Notes in Mathematics 1254, Springer-Verlag, Berlin, 1987. Google Scholar

[G-Sh] [G-Sh] Gelbart, S. and Shahidi, F., Analytic properties of automorphic L-functions. Perspectives in Mathematics 6, Academic Press, Inc., Boston, MA, 1988. Google Scholar

[Ha] [Ha] Harris, M., Occult period invariants and critical values of the degree four L-function of GSp4. In: Contributions to automorphic forms, geometry, and number theory, Johns Hopkins University Press, Baltimore, MD, 2004, 331–354. Google Scholar

[He] [He] Heim, B., Pullbacks of Eisenstein series, Hecke-Jacobi theory and automorphic L-functions. In: Automorphic forms, automorphic representations, and arithmetic, Proc. Sympos. Pure Math. 66, American Mathematical Society, Providence, RI, 1999, pp. 201–238. Google Scholar

[J-1] [J-1] Jacquet, H., Fonctions de Whittaker associées aux groupes de Chevalley. Bull. Soc. Math. France 95(1967), 243–309. Google Scholar

[J-2] [J-2] Jacquet, H., Automorphic forms on GL (2). II. Lecture Notes in Mathematics 278, Springer-Verlag, Berlin-New York, 1972. Google Scholar

[J-S] [J-S] Jacquet, H. and Shalika, J. A., On Euler products and the classification of automorphic representations. I. Amer. J. Math. 103(1981), no. 3, 499–558. Google Scholar

[Ki] [Ki] Kim, H., Automorphic L-functions. In: Lectures on automorphic L -functions, Fields Inst. Monogr. 20, American Mathematical Society, Providence, RI, 2004, pp. 97–201. Google Scholar

[Li] [Li] J.-S. Li, Some results on the unramified principal series of p-adic groups. Math. Ann. 292(1992), no. 4, 747–761. Google Scholar

[Mi-O-1] [Mi-O-1] Miyazaki, T. and Oda, T., Principal series Whittaker functions on Sp(2; ). II. Tohoku Math. J. (2) 50(1998), no. 2, 243–260. Google Scholar

[Mi-O-2] [Mi-O-2] Miyazaki, T. and Oda, T., Errata: Principal series Whittaker functions on Sp(2; ). II. Tohoku Math. J. (2) 54(2002), no. 1, 161–162. Google Scholar

[Mo-1] [Mo-1] Moriyama, T., Spherical functions with respect to the semisimple symmetric pair (Sp (2, ), SL (2, ) × SL (2, ). J. Math. Sci. Univ. Tokyo 6(1999), no. 1, 127–179. Google Scholar

[Mo-2] [Mo-2] Moriyama, T., A remark on Whittaker functions on Sp(2, . J. Math. Sci. Univ. Tokyo 9(2002), no. 4, 627–635. Google Scholar

[Mo-3] [Mo-3] Moriyama, T., Entireness of the spinor L-functions for certain generic cusp forms on GSp (2. Amer. J. Math. 126(2004), no. 4, 899–920. Google Scholar

[Ni] [Ni] Niwa, S., Commutation relations of differential operators and Whittaker functions on Sp 2(. Proc. Japan Acad. Ser. A. Math. Sci. 71(1995), no. 8, 189–191. Google Scholar

[No-1] [No-1] Novodvorsky, M. E., Fonction J pour des groupes orthogonaux. C. R. Acad. Sci. Paris Sér. A-B 280(1975), A1421–A1422. Google Scholar

[No-2] [No-2] Novodvorsky, M. E., Automorphic L-functions for symplectic group GSp (4). In: Automorphic forms, representations and L -functions, Proc. Sympos. Pure Math. 33, American Mathematic Society, Providence, RI, 1979, 87–95. Google Scholar

[O] [O] Oda, T., An explicit integral representation of Whittaker functions on Sp(2; ) for large discrete series representations. Tohoku Math. J. 46(1994), no. 2, 261–279. Google Scholar

[PS] [PS] Piatetski-Shapiro, I. I., L-functions for GSp4, Olga Taussky-Todd: in memoriam. Pacific J. Math. 1997, Special Issue, 259–275. Google Scholar

[PS-S] [PS-S] Piatetski-Shapiro, I. I. and Soudry, D., L and ϵ functions for GSp(4) × GL(2. Proc. Nat. Acad. Sci. U. S. A. 81(1984), no. 12, Phys. Sci., 3924–3927. Google Scholar

[Re] [Re] Reeder, M., p-adic Whittaker functions and vector bundles on flag manifolds. Compositio Math. 85(1993), no. 1, 9–36. Google Scholar

[Ro] [Ro] Rodier, F., Whittaker models for admissible representations of reductive p-adic split groups. In: Harmonic analysis on homogeneous spaces, Proc. Sympos. Pure Math. 26, American Mathematical Society, Providence, RI, 1973, 425–430. Google Scholar

[Sha] [Sha] Shalika, J. A., The multiplicity one theorem for GLn. Ann. of Math. 100(1974), 171–193. Google Scholar

[Shm] [Shm] Shimura, G., The special value of the zeta functions associated with cusp forms. Comm. Pure Appl. Math. 29(1976), no. 6, 783–804. Google Scholar

[So-1] [So-1] Soudry, D., The L and γ factors for generic representations of GSp (4, k ) × GL (2, k ) over a local non-Archimedean field k. Duke Math. J. 51(1984), no. 2, 355–394. Google Scholar

[So-2] [So-2] Soudry, D., On the Archimedean theory of Rankin-Selberg convolutions for SO2l +1 × GLn. Ann. Sci. École Norm. Sup. 28(1995), no. 2, 161–224. Google Scholar

[Wa] [Wa] Wallach, N. R., Asymptotic expansions of generalized matrix entries of representations of real reductive groups. In: Lie group representations, I., Lecture Notes in Math. 1024, Springer, Berlin, 1983, pp. 287–369. Google Scholar

[W-W] [W-W] Whittaker, E. T. and Watson, G. N., A course of modern analysis. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principla transcendental functions. Reprint of the fourth (1927) edition, Cambridge University Press, Cambrigde, 1996. Google Scholar