Orbital $L$ -functions for the Space of Binary Cubic Forms

Taniguchi, Takashi ; Thorne, Frank

Canadian journal of mathematics, Tome 65 (2013) no. 6, pp. 1320-1383

Résumé

We introduce the notion of orbital $L$ -functions for the space of binary cubic forms and investigate their analytic properties. We study their functional equations and residue formulas in some detail. Aside from their intrinsic interest, the results from this paper are used to prove the existence of secondary terms in counting functions for cubic fields. This is worked out in a companion paper.

Détail

DOI : 10.4153/CJM-2013-027-0

Mots-clés : 11M41, 11E76, binary cubic forms, prehomogeneous vector spaces, Shintani zeta functions, L-functions, cubic rings and fields

Bibliographie

[1] [1] Bhargava, M., Shankar, A., and Tsimerman, J., On the Davenport–Heilbronn theorem and second order terms. Invent. Math. 193(2013), 439–499. Google Scholar | DOI

[2] [2] Datskovsky, B. and Wright, D. J., The adelic zeta function associated with the space of binary cubic forms II: Local theory. J. Reine Angew. Math. 367(1986), 27–75. Google Scholar

[3] [3] Datskovsky, B., Density of discriminants of cubic extensions. J. Reine Angew. Math. 386(1988), 116–138. Google Scholar

[4] [4] Davenport, H. and Heilbronn, H., On the density of dsicriminants of cubic fields. II. Proc. Roy. Soc. London Ser. A 322(1971), no. 1551, 405–420. Google Scholar

[5] [5] Delone, B. N. and D. K. Faddeev, , The theory of irrationalities of the third degree. Transl. Math. Monogr. 10, American Mathematical Society, Providence, 1964. Google Scholar

[6] [6] Denef, J. and Gyoja, A., Character sums associated to prehomogeneous vector spaces. Compositio Math. 113(1998), 273–346. Google Scholar | DOI

[7] [7] Fouvry, E. and Katz, N., A general stratification theorem for exponential sums, and applications. J. Reine Angew. Math. 540(2001), 115–166. Google Scholar

[8] [8] Gan, W. T., Gross, B., and Savin, G., Fourier coefficients of modular forms on G. Duke Math. J. 115(2002), 105–169. Google Scholar | DOI

[9] [9] Gelbart, S., Automorphic Forms on Adele Groups. Princeton University Press, Princeton, 1975. Google Scholar

[10] [10] Ibukiyama, T. and Saito, H., On zeta functions associated to symmetric matrices and an explicit conjecture on dimensions of Siegel modular forms of general degree. Internat. Math. Res. Notices 8(1992), 161–169. Google Scholar

[11] [11] Ibukiyama, T. and Saito, H., On zeta functions associated to symmetric matrices III: An explicit form of L-functions. Nagoya Math. J. 146(1997), 149–183. Google Scholar

[12] [12] Iwaniec, H. and Kowalski, E., Analytic Number Theory. Amer. Math. Soc. Colloq. Publ. , Amer. Math. Soc., Providence, Rhode Island, 2004. Google Scholar

[13] [13] Jones, J. and Roberts, D., A database of local fields. J. Symbolic Comput. 41(2006), 80–97. Accompanying database available online at http://math.la.asu.edu/_jj/localfields/. Google Scholar | DOI

[14] [14] Mori, S., Orbital Gauss sums for the space of binary cubic forms over a finite field. In preparation. Google Scholar

[15] [15] Nakagawa, J., On the relations among the class numbers of binary cubic forms. Invent. Math. 134(1998), 101–138. Google Scholar | DOI

[16] [16] Ohno, Y., A conjecture on coincidence among the zeta functions associated with the space of binary cubic forms. Amer. J. Math. 119(1997), 1083–1094. Google Scholar | DOI

[17] [17] Ohno, Y. and Taniguchi, T., Relations among Dirichlet series whose coefficients are class numbers of binary cubic forms II.Preprint, 2009. arxiv:1112.5029 Google Scholar

[18] [18] Ohno, Y., Taniguchi, T., and Wakatsuki, S., Relations among Dirichlet series whose coefficients are class numbers of binary cubic forms. Amer. J. Math. 131(2009), 1525–1541. Google Scholar | DOI

[19] [19] PARI/GP. version 2.3.4, Bordeaux, 2008. Available from http://pari.math.u-bordeaux.fr/. Google Scholar

[20] [20] Saito, H., A generalization of Gauss sums and its applications to Siegel modular forms and L-functions associated with the vector space of quadratic forms. J. Reine Angew. Math. 416(1991), 91–142. Google Scholar

[21] [21] Saito, H., On L-functions associated with the vector space of binary quadratic forms. Nagoya Math. J. 130(1993), 149–176. Google Scholar

[22] [22] Sato, F., On functional equations of zeta distributions. Adv. Studies in Pure Math. 15(1989), 465–508. Google Scholar

[23] [23] Sato, M. and Kimura, T., A classification of irreducible prehomogeneous vector spaces and their relative invariants. Nagoya Math. J. 65(1977), 1–155. Google Scholar

[24] [24] Sato, M. and Shintani, T., On zeta functions associated with prehomogeneous vector spaces. Ann. of Math. 100(1974), 131–170. Google Scholar | DOI

[25] [25] Shintani, T., On Dirichlet series whose coefficients are class-numbers of integral binary cubic forms. J. Math. Soc. Japan 24(1972), 132–188. Google Scholar | DOI

[26] [26] Shintani, T., On zeta-functions associated with vector spaces of quadratic forms. J. Fac. Sci. Univ. Tokyo, Sect. IA 22(1975), 25–66. Google Scholar

[27] [27] Taniguchi, T., On the zeta functions of prehomogeneous vector spaces for a pair of simple algebras. Ann. Inst. Fourier 57(2007), 1331–1358. Google Scholar | DOI

[28] [28] Taniguchi, T. and Thorne, F., Secondary terms in counting functions for cubic fields. Duke Math. J., to appear. Google Scholar

[29] [29] Wright, D. J., The adelic zeta function associated to the space of binary cubic forms part I: Global theory. Math. Ann. 270(1985), 503–534. Google Scholar | DOI

[30] [30] Wright, D. J., Cubic character sums of cubic polynomials. Proc. Amer. Math. Soc. 100(1987), 409–413. Google Scholar | DOI

[31] [31] Wright, D. J. and Yukie, A., Prehomogeneous vector spaces and field extensions. Invent. Math. 110(1992), 283–314. Google Scholar | DOI

[32] [32] Yukie, A., Shintani Zeta Functions. London Math. Soc. Lecture Note Ser. 183, Cambridge University Press, Cambridge, 1993. Google Scholar