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The adelic zeta function associated with the space of binary cubic forms with coefficients in a function field


Author: Boris A. Datskovsky
Journal: Trans. Amer. Math. Soc. 299 (1987), 719-745
MSC: Primary 11E76
MathSciNet review: 869230
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Abstract: In this paper we study the adelic zeta function associated with the prehomogeneous vector space of binary cubic forms, defined over a function field. We establish its rationality, find its poles and residues and a simple functional equation that this zeta function satisfies.


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  • [1] F. Arndt. Zur Theorie der binaren kubischen Formen, J. Reine Angew. Math. 53 (1857), 309-321.
  • [2] Armand Borel, Some finiteness properties of adele groups over number fields, Inst. Hautes Études Sci. Publ. Math. 16 (1963), 5–30. MR 0202718
  • [3] B. Datskovsky, On zeta functions associated with the space of binary cubic forms with coefficients in a function field, Ph.D. thesis, Harvard Univ., 1984.
  • [4] J. W. Cogdell, Congruence zeta functions for 𝑀₂(𝑄) and their associated modular forms, Math. Ann. 266 (1983), no. 2, 141–198. MR 724736, 10.1007/BF01458441
  • [5] H. Davenport, On the class-number of binary cubic forms. I, J. London Math. Soc. 26 (1951), 183–192. MR 0043822
  • [6] H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields, Bull. London Math. Soc. 1 (1969), 345–348. MR 0254010
  • [7] H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields. II, Proc. Roy. Soc. London Ser. A 322 (1971), no. 1551, 405–420. MR 0491593
  • [8] B. N. Delone and D. K. Faddeev, The theory of irrationalities of the third degree, Translations of Mathematical Monographs, Vol. 10, American Mathematical Society, Providence, R.I., 1964. MR 0160744
  • [9] G. Eisenstein, Untersuchungen über die cubischen Formen mit zwei Variabeln, J. Reine Angew. Math. 27 (1844), 89-104.
  • [10] G. Harder, Minkowskische Reduktionstheorie über Funktionenkörpern, Invent. Math. 7 (1969), 33–54 (German). MR 0284441
  • [11] Stephen S. Gelbart, Automorphic forms on adèle groups, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1975. Annals of Mathematics Studies, No. 83. MR 0379375
  • [12] Roger Godement and Hervé Jacquet, Zeta functions of simple algebras, Lecture Notes in Mathematics, Vol. 260, Springer-Verlag, Berlin-New York, 1972. MR 0342495
  • [13] Jun-ichi Igusa, On certain representations of semi-simple algebraic groups and the arithmetic of the corresponding invariants. I, Invent. Math. 12 (1971), 62–94. MR 0297771
  • [14] Jun-ichi Igusa, Forms of higher degree, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 59, Tata Institute of Fundamental Research, Bombay; by the Narosa Publishing House, New Delhi, 1978. MR 546292
  • [15] Jun-ichi Igusa, Some results on 𝑝-adic complex powers, Amer. J. Math. 106 (1984), no. 5, 1013–1032. MR 761577, 10.2307/2374271
  • [16] Tomio Kubota, Elementary theory of Eisenstein series, Kodansha Ltd., Tokyo; Halsted Press [John Wiley & Sons], New York-London-Sydney, 1973. MR 0429749
  • [17] R. P. Langlands, Eisenstein series, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) Amer. Math. Soc., Providence, R.I., 1966, pp. 235–252. MR 0249539
  • [18] Robert P. Langlands, On the functional equations satisfied by Eisenstein series, Lecture Notes in Mathematics, Vol. 544, Springer-Verlag, Berlin-New York, 1976. MR 0579181
  • [19] Wen Ch’ing Winnie Li, On modular functions in characteristic 𝑝, Trans. Amer. Math. Soc. 246 (1978), 231–259. MR 515538, 10.1090/S0002-9947-1978-0515538-9
  • [20] Wen Ch’ing Winnie Li, Eisenstein series and decomposition theory over function fields, Math. Ann. 240 (1979), no. 2, 115–139. MR 524661, 10.1007/BF01364628
  • [21] M. Muro, Microlocal analysis and the calculations of functional equations and residues of zeta functions associated with the vector spaces of quadratic forms (preprint).
  • [22] Fumihiro Satō, Zeta functions in several variables associated with prehomogeneous vector spaces. I. Functional equations, Tôhoku Math. J. (2) 34 (1982), no. 3, 437–483. MR 676121, 10.2748/tmj/1178229205
  • [23] M. Sato, M. Kashiwara, T. Kimura, and T. Ōshima, Microlocal analysis of prehomogeneous vector spaces, Invent. Math. 62 (1980/81), no. 1, 117–179. MR 595585, 10.1007/BF01391666
  • [24] Mikio Sato and Takuro Shintani, On zeta functions associated with prehomogeneous vector spaces, Ann. of Math. (2) 100 (1974), 131–170. MR 0344230
  • [25] Takuro Shintani, On Dirichlet series whose coefficients are class numbers of integral binary cubic forms, J. Math. Soc. Japan 24 (1972), 132–188. MR 0289428
  • [26] Takuro Shintani, On zeta-functions associated with the vector space of quadratic forms, J. Fac. Sci. Univ. Tokyo Sect. I A Math. 22 (1975), 25–65. MR 0384717
  • [27] Toshiaki Suzuki, On zeta functions associated with quadratic forms of variable coefficients, Nagoya Math. J. 73 (1979), 117–147. MR 524011
  • [28] J. Tate, Fourier analysis in number fields and Hecke's zeta function, Ph.D. thesis, Princeton Univ., 1950.
  • [29] André Weil, Adeles and algebraic groups, Progress in Mathematics, vol. 23, Birkhäuser, Boston, Mass., 1982. With appendices by M. Demazure and Takashi Ono. MR 670072
  • [30] André Weil, Sur certains groupes d’opérateurs unitaires, Acta Math. 111 (1964), 143–211 (French). MR 0165033
  • [31] André Weil, Sur la formule de Siegel dans la théorie des groupes classiques, Acta Math. 113 (1965), 1–87 (French). MR 0223373
  • [32] -, Fonction zêta et distributions, Séminaire Bourbaki, No. 316, 1966; Math. Ann. 168 (1967), 140-156.
  • [33] André Weil, Basic number theory, 3rd ed., Springer-Verlag, New York-Berlin, 1974. Die Grundlehren der Mathematischen Wissenschaften, Band 144. MR 0427267
  • [34] -, On the analogue of the modular group in characteristic $ p$, Proc. Conf. in honor of M. Stone, Springer-Verlag, Berlin and New York, 1970, pp. 211-223.
  • [35] D. Wright, Dirichlet series associated with the space of binary cubic forms with coefficients in a number field, Ph.D. Thesis, Harvard Univ., 1982.
  • [36] David J. Wright, The adelic zeta function associated to the space of binary cubic forms. I. Global theory, Math. Ann. 270 (1985), no. 4, 503–534. MR 776169, 10.1007/BF01455301

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DOI: https://doi.org/10.1090/S0002-9947-1987-0869230-7
Article copyright: © Copyright 1987 American Mathematical Society