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A construction of quintic rings

Published online by Cambridge University Press:  22 January 2016

Anthony C. Kable  and
Akihiko Yukie
Anthony C. Kable
Affiliation:
Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, USA, [email protected]
Akihiko Yukie
Affiliation:
Mathematical Institute, Tôhoku University, Sendai, 980-8578, Japan, [email protected]
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Abstract

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We construct a discriminant-preserving map from the set of orbits in the space of quadruples of quinary alternating forms over the integers to the set of isomorphism classes of quintic rings. This map may be regarded as an analogue of the famous map from the set of equivalence classes of integral binary cubic forms to the set of isomorphism classes of cubic rings and may be expected to have similar applications. We show that the ring of integers of every quintic number field lies in the image of the map. These results have been used to establish an upper bound on the number of quintic number fields with bounded discriminant.

Keywords

11R0411R2115A7214L30
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 173 , 2004 , pp. 163 - 203
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2004

References

[1] Amano, K., Fujigami, M. and Kogiso, T., Construction of Irreducible Relative Invariant of the Prehomogeneous Vector Space (SL5 × GL4,Λ2(ℂ5)⊗ℂ4), Linear Algebra Appl., 355 (2002), 215–222.Google Scholar
[2] Bhargava, M., Higher Composition Laws, Ph.D. Thesis, Princeton University, 2001.Google Scholar
[3] Bhargava, M., Gauss Composition and Generalizations, Algorithmic Number Theory: 5th International Symposium (Fieker, C. and Kohel, D. R., eds.), Springer Lecture Notes in Computer Science, vol. 2369, Springer, New York (2002), pp. 1–8.Google Scholar
[4] Cayley, A., On the theory of linear transformations, The Collected Mathematical Papers of Arthur Cayley, vol. 1, Cambridge University Press, Cambridge (1889), pp. 80–94.Google Scholar
[5] Cohen, H., Diaz, F. Diaz y and Olivier, M., Counting discriminants of number fields of degree up to four, Algorithmic Number Theory (Leiden 2000), Lecture Notes in Computer Science, vol. 1838, Springer, Berlin (2000), pp. 269–283.Google Scholar
[6] Curtis, C. W. and Reiner, I., Methods of Representation Theory with Applications to Finite Groups and Integral Orders, Vol. 1, Wiley Classics Library, John Wiley & Sons, New York, 1990.Google Scholar
[7] Davenport, H. and Heilbronn, H., On the density of discriminants of cubic fields, Bull. Lond. Math. Soc., 1 (1969), 345–348.Google Scholar
[8] Davenport, H. and Heilbronn, H., On the density of the discriminants of cubic fields. II, Proc. Roy. Soc. London Ser. A. 322 (1971), 405–420.Google Scholar
[9] Delone, B. N. and Fadeev, D. K., Theory of Irrationalities of the Third Degree, Translations of Mathematical Monographs, Vol. 10, American Mathematical Society, Providence, 1964.Google Scholar
[10] Dieudonné, J., Sur la reduction canonique des couples de matrices, Bull. Soc. math. France, 74 (1946), 130–146.Google Scholar
[11] Gel’fand, I. M., Kapranov, M. M. and Zelevinsky, A. V., Discriminants, Resultants and Multidimensional Determinants, Mathematics: Theory and Applications, Birkhäuser, Boston, 1994.Google Scholar
[12] Gan, W. T., Gross, B. and Savin, G., Fourier Coefficients of Modular Forms on G2 , preprint (2001).Google Scholar
[13] Gurevich, G. B., Foundations of the Theory of Algebraic Invariants, translated by Radok, J. R. M. and Spencer, A. J. M., Noordhoff, P., Groningen, 1964.Google Scholar
[14] Gyoja, A. and Omoda, Y., Characteristic cycles of certain character sheaves, Indag. Math. (N.S.), 12 (2001), 329–335.Google Scholar
[15] Kable, A. C., Classes of integral 3-tensors on 2-space, Mathematika, 47 (2000), 205–217.Google Scholar
[16] Kable, A. C., The concomitants of a prehomogeneous vector space, to appear in J. Algebra.Google Scholar
[17] Kable, A. C. and Yukie, A., Prehomogeneous vector spaces and field extensions II, Invent. Math., 130 (1997), 315–344.Google Scholar
[18] Kable, A. C. and Yukie, A., On the Number of Quintic Fields, preprint (2002).Google Scholar
[19] Kawanaka, N., Generalized Gel’fand-Graev representations of exceptional simple algebraic groups over a finite field I, Invent. math., 84 (1986), 575–616.Google Scholar
[20] Kimura, T., Sato, F. and Zhu, X.-W., On the poles of p-adic complex powers and the b-functions of prehomogeneous vector spaces, Amer. J. Math., 112 (1990), 423–437.Google Scholar
[21] Kimura, T. and Sato, M., A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J., 65 (1977), 1–155.Google Scholar
[22] Knop, F. and Menzel, G., Duale Varietäten von Fahnenvarietäten, Comment. Math. Helv., 62 (1987), 38–61.Google Scholar
[23] Lusztig, G., Introduction to character sheaves, The Arcata Conference on Representations of Finite Groups, Proc. Sympos. Pure Math., vol. 47, part 1, Amer. Math. Soc., Providence (1987), pp. 165–179.Google Scholar
[24] Muro, M., Sato, M. and Shintani, T., Theory of prehomogeneous vector spaces (algebraic part) – the English translation of Sato’s lecture from Shintani’s note, Nagoya Math. J., 120 (1990), 1–34.Google Scholar
[25] Narkiewicz, W., Elementary and Analytic Theory of Algebraic Numbers, PWN, Warsaw, 1974.Google Scholar
[26] Ozeki, I., On the microlocal structure of the regular prehomogeneous vector space associated with SL(5) × GL(4) I, Proc. Japan Acad. 55, Ser. A (1979), 37–40.Google Scholar
[27] Ozeki, I., On the microlocal structure of the regular prehomogeneous vector space associated with SL(5) × GL(4) I, Publ. Res. Inst. Math. Sci., 26 (1990), no. 3, 539–584.Google Scholar
[28] Scharlau, R., Paare alternierender Formen, Math. Z., 147 (1976), 13–19.Google Scholar
[29] Schmidt, W., Number fields of given degree and bounded discriminant, Columbia University Number Theory Seminar (New York 1992), Astérisque, 228 (1995), 189–195.Google Scholar
[30] Waterloo Maple Inc., “Maple 7”, copyright 2001.Google Scholar
[31] Witte, D., Yukie, A. and Zierau, R., Prehomogeneous vector spaces and ergodic theory II, Trans. Amer. Math. Soc., 352 (2000), 1687–1708.Google Scholar
[32] Wright, D. and Yukie, A., Prehomogeneous vector spaces and field extensions, Invent. math., 110 (1992), 283–314.Google Scholar
[33] Yukie, A., Density theorems related to prehomogeneous vector spaces, Automorphic forms, automorphic representations and automorphic L-functions over algebraic groups, Sūrikaisekikenkyūsho Kōkyūroku, 1173 (2000), 171–183.Google Scholar
[34] Yukie, A., Shintani Zeta Functions, Lond. Math. Soc. Lecture Notes, Vol. 183, Cambridge UP, Cambridge, 1993.Google Scholar