Values of Twisted Tensor L-functions of Automorphic Forms Over Imaginary Quadratic Fields

Lanphier, Dominic ; Skogman, Howard ; Ochiai, Hiroyuki

Canadian journal of mathematics, Tome 66 (2014) no. 5, pp. 1078-1109

Résumé

Let $K$ be a complex quadratic extension of $\mathbb{Q}$ and let ${{\mathbb{A}}_{K}}$ denote the adeles of $K$ . We find special values at all of the critical points of twisted tensor $L$ -functions attached to cohomological cuspforms on $G{{L}_{2}}\left( {{\mathbb{A}}_{K}} \right)$ and establish Galois equivariance of the values. To investigate the values, we determine the archimedean factors of a class of integral representations of these $L$ -functions, thus proving a conjecture due to Ghate. We also investigate analytic properties of these $L$ -functions, such as their functional equations.

Détail

DOI : 10.4153/CJM-2013-047-5

Mots-clés : 11F67, 11F37, twisted tensor L-function, cuspform, hypergeometric series

Bibliographie

[1] Andrews, G. E., Askey, R., and Roy, R., Special functions. Encyclopedia of Mathematics and its Applications, 71, Cambridge Univeristy Press, Cambridge, 1999. Google Scholar

[2] Asai, T., On certain Dirichlet series associated with Hilbert modular forms and Rankin's method. Math. Ann. 226(1977), no. 1, 81–94. Google Scholar | DOI

[3] Deligne, P., Valeurs de fonctions Let priodes d'intègrales. In: Automorphic forms, automorphic representations, and L-Functions (Proc. Symp. Pure Math. Oregon State Univ., Corvallis, Ore, 1977), part 2, American Mathematical Society, Providence, RI, 1979, pp. 313–346. Google Scholar

[4] Flicker, Y. Z., On zeroes of the twisted tensor L-function. Math. Ann. 297(1993), no. 2, 199–219. Google Scholar | DOI

[5] Flicker, Y .Z. and Zinoviev, D., On poles of twisted tensor L-functions. Proc. Japan Acad. Ser. A Math. Sci. 71(1995), no. 6, 114–116. Google Scholar | DOI

[6] Ghate, E., Critical Values of the Asai L-function in the imaginary quadratic case. Thesis (Ph.D.), University of California, Los Angeles, Proquest LLC, Ann Arbor, MI, 1996. Google Scholar

[7] Ghate, E., Critical values of the twisted tensor L-function in the imaginary quadratic case. Duke Math J. 96(1999), no. 3, 595–638. Google Scholar | DOI

[8] Ghate, E., Critical values of twisted tensor L-functions over CM-fields. In: Automorphic forms, automorphic representations, and arithmetic (FortWorth, TX, 1996), Proc. Symp. Pure Math., 66, Part 1, American Mathematical Society, Providence, RI, 1999, pp. 87–109. Google Scholar

[9] Harder, G., Eisenstein cohomology of arithmetic groups. The case GL2. Invent. Math. 89(1987), no. 1, 37–118. Google Scholar | DOI

[10] Hida, H., p-Ordinary cohomology groups for SL(2) over number fields. Duke Math. J. 69(1993), no. 2,259–314. Google Scholar | DOI

[11] Hida, H., On the critical values of L-functions of GL(2) and GL(2)✗ GL(2). Duke Math. J. 74(1994), no. 2, 431–528. Google Scholar | DOI

[12] Knuth, D. E., The art of computer programming. Addison-Wesley, Reading, Mass, 1968. Google Scholar

[13] Krishnamurthy, M., The Asai transfer to GL4 via the Langlands-Shahidi method. Int. Math. Res. Not. 2003, no. 41, 2221–2254. Google Scholar

[14] Mahnkopf, J., Cohomology of arithmetic groups, parabolic subgroups and special values of L-functions of GLn. J. Inst. Math. Jussieu 4(2005), no. 4, 553–637. Google Scholar | DOI

[15] Miyake, T., Modular forms. Springer-Verlag, Berlin, 1989. Google Scholar

[16] Raghuram, A. and Shahidi, F., On certain period relations for cusp forms on GLn. Int. Math. Res. Not. IMRN 2008, Art. ID rnn 077, 23 pp. Google Scholar

[17] Shimura, G., The critical values of certain Dirichlet series attached to Hilbert modular forms. Duke Math. J. 63(1991), no. 3, 557–613. Google Scholar | DOI

[18] Shimura, G., On modular forms of half-integral weight. Ann. Math. 97(1973), 440–481. Google Scholar | DOI

[19] Takase, K., On certain Dirichlet series associated with automorphic forms on SL(2; C). Manuscripta Math. 56(1986), no. 3, 293–312. Google Scholar | DOI

[20] Zhao, Y., Certain Dirichlet series attached to automorphic forms over imaginary quadratic fields. Duke Math. J. 72(1993), no. 3, 695–724. Google Scholar | DOI