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Shintani's zeta function is not a finite sum of Euler products


Author: Frank Thorne
Journal: Proc. Amer. Math. Soc. 142 (2014), 1943-1952
MSC (2010): Primary 11M41, 11R16
DOI: https://doi.org/10.1090/S0002-9939-2014-12064-8
Published electronically: March 5, 2014
MathSciNet review: 3182013
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Abstract: We prove that the Shintani zeta function associated to the space of binary cubic forms cannot be written as a finite sum of Euler products. Our proof also extends to several closely related Dirichlet series. This answers a question of Wright in the negative.


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  • [1] Karim Belabas, On quadratic fields with large 3-rank, Math. Comp. 73 (2004), no. 248, 2061-2074. MR 2059751 (2005c:11132), https://doi.org/10.1090/S0025-5718-04-01632-1
  • [2] M. Bhargava, A. Shankar, and J. Tsimerman, On the Davenport-Heilbronn theorem and second order terms, Invent. Math. 193 (2013), no. 2, 439-499.MR 3090184
  • [3] E. Bombieri and D. A. Hejhal, On the distribution of zeros of linear combinations of Euler products, Duke Math. J. 80 (1995), no. 3, 821-862. MR 1370117 (96m:11071), https://doi.org/10.1215/S0012-7094-95-08028-4
  • [4] Henri Cohen and Anna Morra, Counting cubic extensions with given quadratic resolvent, J. Algebra 325 (2011), 461-478. MR 2745550 (2012b:11168), https://doi.org/10.1016/j.jalgebra.2010.08.027
  • [5] H. Cohen and F. Thorne, Dirichlet series associated to cubic fields with given quadratic resolvent. To appear in Michigan Math. J.
  • [6] Boris Datskovsky and David J. Wright, The adelic zeta function associated to the space of binary cubic forms. II. Local theory, J. Reine Angew. Math. 367 (1986), 27-75. MR 839123 (87m:11034), https://doi.org/10.1515/crll.1986.367.27
  • [7] Boris Datskovsky and David J. Wright, Density of discriminants of cubic extensions, J. Reine Angew. Math. 386 (1988), 116-138. MR 936994 (90b:11112), https://doi.org/10.1515/crll.1988.386.116
  • [8] 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 (58 #10816)
  • [9] B. Hough, Equidistribution of Heegner points associated to the 3-part of the class group, preprint.
  • [10] Tomoyoshi Ibukiyama and Hiroshi Saito, On zeta functions associated to symmetric matrices. I. An explicit form of zeta functions, Amer. J. Math. 117 (1995), no. 5, 1097-1155. MR 1350594 (96j:11120), https://doi.org/10.2307/2374973
  • [11] A. Morra, Comptage asymptotique et algorithmique d'extensions cubiques relatives (in English), thesis, Université Bordeaux I, 2009. Available online at http://perso.univ-rennes1.fr/anna.morra/these.pdf.
  • [12] Jin Nakagawa, On the relations among the class numbers of binary cubic forms, Invent. Math. 134 (1998), no. 1, 101-138. MR 1646578 (99j:11036), https://doi.org/10.1007/s002220050259
  • [13] Jin Nakagawa and Kuniaki Horie, Elliptic curves with no rational points, Proc. Amer. Math. Soc. 104 (1988), no. 1, 20-24. MR 958035 (89k:11113), https://doi.org/10.2307/2047452
  • [14] Yasuo Ohno, A conjecture on coincidence among the zeta functions associated with the space of binary cubic forms, Amer. J. Math. 119 (1997), no. 5, 1083-1094. MR 1473069 (98k:11037)
  • [15] PARI/GP, version 2.3.4, Bordeaux, 2008, available from http://pari.math.u-bordeaux.fr/.
  • [16] 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 (44 #6619)
  • [17] J. Tate, Fourier analysis in number fields and Hecke's zeta-functions, thesis (Princeton, 1950); reprinted in J.W.S. Cassels and A. Fröhlich, eds., Algebraic number theory, Academic Press, London, 1986.MR 0217026 (36 #121)
  • [18] T. Taniguchi and F. Thorne, Secondary terms in counting functions for cubic fields, Duke Math. J. 162 (2013), no. 13, 2451-2508.MR 3127806
  • [19] T. Taniguchi and F. Thorne, Orbital $ L$-functions for the space of binary cubic forms, Canad. J. Math. 65 (2013), no. 6, 1320-1383. MR 3121674
  • [20] F. Thorne, Analytic properties of Shintani zeta functions, Proceedings of the RIMS Symposium on automorphic forms, automorphic representations, and related topics, Kyoto, 2010. (Also available from the author's website.)
  • [21] 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 (86k:11023), https://doi.org/10.1007/BF01455301
  • [22] David J. Wright, Twists of the Iwasawa-Tate zeta function, Math. Z. 200 (1989), no. 2, 209-231. MR 978295 (90c:11087), https://doi.org/10.1007/BF01230282
  • [23] Akihiko Yukie, Shintani zeta functions, London Mathematical Society Lecture Note Series, vol. 183, Cambridge University Press, Cambridge, 1993. MR 1267735 (95h:11037)
  • [24] Y. Zhao, doctoral thesis, University of Wisconsin, in preparation.

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Additional Information

Frank Thorne
Affiliation: Department of Mathematics, University of South Carolina, 1523 Greene Street, Columbia, South Carolina 29201
Email: thorne@math.sc.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12064-8
Received by editor(s): July 11, 2012
Published electronically: March 5, 2014
Communicated by: Ken Ono
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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