Hostname: page-component-89b8bd64d-n8gtw Total loading time: 0 Render date: 2026-05-09T13:16:12.812Z Has data issue: false hasContentIssue false
  • English
  • Français

An Explicit Treatment of Cubic Function Fields with Applications

Published online by Cambridge University Press:  20 November 2018

E. Landquist ,
P. Rozenhart ,
R. Scheidler ,
J. Webster  and
Q. Wu
E. Landquist
Affiliation:
Department of Mathematics, Kutztown University of Pennsylvania, Kutztown, PA 19530, USA, e-mail: elandqui@kutztown.edu
P. Rozenhart
Affiliation:
Department of Mathematics and Statistics, University of Calgary, Calgary, AB T2N 1N4, e-mail: pieter@math.ucalgary.ca, rscheidl@math.ucalgary.ca, quwu@math.ucalgary.ca
R. Scheidler
Affiliation:
Department of Mathematics and Statistics, University of Calgary, Calgary, AB T2N 1N4, e-mail: pieter@math.ucalgary.ca, rscheidl@math.ucalgary.ca, quwu@math.ucalgary.ca
J. Webster
Affiliation:
Department of Mathematics and Statistics, University of Calgary, Calgary, AB T2N 1N4, e-mail: pieter@math.ucalgary.ca, rscheidl@math.ucalgary.ca, quwu@math.ucalgary.ca
Q. Wu
Affiliation:
Department of Mathematics, Bates College, Lewiston, ME 04240, USA, e-mail: jwebster@bates.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We give an explicit treatment of cubic function fields of characteristic at least five. This includes an efficient technique for converting such a field into standard form, formulae for the field discriminant and the genus, simple necessary and sufficient criteria for non-singularity of the defining curve, and a characterization of all triangular integral bases. Our main result is a description of the signature of any rational place in a cubic extension that involves only the defining curve and the order of the base field. All these quantities only require simple polynomial arithmetic as well as a few square-free polynomial factorizations and, in some cases, square and cube root extraction modulo an irreducible polynomial. We also illustrate why and how signature computation plays an important role in computing the class number of the function field. This in turn has applications to the study of zeros of zeta functions of function fields.

Keywords

14H0511R5814H4511G2011G3011R1611R29cubic function fielddiscriminantnon-singularityintegral basisgenussignature of a placeclass number

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 62 , Issue 4 , 01 August 2010 , pp. 787 - 807
Copyright
Copyright © Canadian Mathematical Society 2010

References

[1] Albert, A. A., A determination of the integers of all cubic fields. Ann. of Math. 31(1930), no. 4, 550–566. doi:10.2307/1968153Google Scholar
[2] Bauer, M. L., The arithmetic of certain cubic function fields. Math. Comp. 73(2004), no. 245, 387–413. (electronic) doi:10.1090/S0025-5718-03-01559-XGoogle Scholar
[3] Buchmann, J. A. and H.W. Lenstra, Jr., Approximating rings of integers in number fields. J. Théor. Nombres Bordeaux 6(1994), no. 2, 221–260.Google Scholar
[4] Cantor, D. G., Computing in the Jacobian of a hyperelliptic curve. Math. Comp. 48(1987), no. 177, 95–101. doi:10.2307/2007876Google Scholar
[5] Chistov, A. L., The complexity of constructing the ring of integers in a global field. Soviet. Math. Dokl. 39(1989), no. 5, 597–600.Google Scholar
[6] Cohen, H., A Course in Computational Algebraic Number Theory. Graduate Texts in Mathematics 138. Springer-Verlag, Berlin, 1993.Google Scholar
[7] Delone, B. N. and Faddeev, K., The Theory of Irrationalities of the Third Degree. Translations of Mathematical Monographs 10. American Mathematical Society, Providence, RI, 1964.Google Scholar
[8] Hasse, H., Number Theory. Classics in Mathematics, Springer-Verlag, Berlin, 2002.Google Scholar
[9] Hess, F., Computing Riemann-Roch spaces in algebraic function fields and related topics. J. Symbolic Comput. 33(2002), no. 4, 425–445. doi:10.1006/jsco.2001.0513Google Scholar
[10] Katz, N. M. and Sarnak, P.. Random Matrices, Frobenius Eigenvalues and Monodromy. American Mathematical Society Colloquium Publications 45. Amererican Mathematical Society, Providence, RI, 1999.Google Scholar
[11] Katz, N. M. and Sarnak, P., Zeroes of zeta functions and symmetry. Bull. Amer. Math. Soc. 36(1999), no. 1, 1–26. doi:10.1090/S0273-0979-99-00766-1Google Scholar
[12] Khuri-Makdisi, K., Linear algebra algorithms for divisors on an algebraic curve. Math. Comp. 73(2004), no. 245, 333–357. (electronic) doi:10.1090/S0025-5718-03-01567-9Google Scholar
[13] Lee, Y., The unit rank classification of a cubic function field by its discriminant. Manuscripta Math. 116(2005), no. 2, 173–181. doi:10.1007/s00229-004-0530-5Google Scholar
[14] Lee, Y., Scheidler, R., and Yarrish, C., Computation of the fundamental units and the regulator of a cyclic cubic function field. Experiment. Math. 12(2003), no. 2, 211–225.Google Scholar
[15] Lehmer, D. H., Computer technology applied to the theory of numbers. In: Studies in Number Theory. Math. Assoc. Amer., 1969, pp. 117–151.Google Scholar
[16] Llorente, P. and Nart, E., Effective determination of the decomposition of the rational primes in a cubic field. Proc. Amer. Math. Soc. 87(1983), no. 4, 579–585. doi:10.2307/2043339Google Scholar
[17] Lorenzini, D., An Invitation to Arithmetic Geometry. Graduate Studies in Mathematics 9. American Mathematical Society, Providence, RI, 1996.Google Scholar
[18] Pohst, M. and Zassenhaus, H., Algorithmic Algebraic Number Theory. Revised reprint. Encyclopedia of Mathematics and its Applications 30. Cambridge University Press, Cambridge, 1997.Google Scholar
[19] Rosen, M., Number Theory in Function Fields. Graduate Texts in Mathematics 210. Springer-Verlag, New York, 2002.Google Scholar
[20] Scheidler, R., Ideal arithmetic and infrastructure in purely cubic function fields. J. Théorie Nombres Bordeaux, 13(2001), no. 2, 609–631 Google Scholar
[21] Scheidler, R., Algorithmic aspects of cubic function fields. In: Algorithmic Number Theory. Lecture Notes in Comput. Sci. 3076. Springer-Verlag, Berlin, 2004, pp. 395–410.Google Scholar
[22] Scheidler, R. and Stein, A., Approximating Euler products and class number computation in algebraic function fields. To appear in Rocky Mountain J. Math.Google Scholar
[23] Scheidler, R. and Stein, A., Voronoi's algorithm in purely cubic congruence function fields of unit rank 1. Math. Comp. 69(2000), no. 231, 1245–1266. doi:10.1090/S0025-5718-99-01136-9Google Scholar
[24] Scheidler, R. and Stein, A., Class number approximation in cubic function fields. Contr. Discrete Math. 2(2007), no. 2, 107–132. (electronic)Google Scholar
[25] Shanks, D.. Five number-theoretic algorithms. In: Proc. Second Manitoba Conference on Numerical Mathematics. Congres 1973. Utilitas Math., Winnipeg, 1973, pp. 51–70.Google Scholar
[26] Stein, A. and Teske, E., Explicit bounds and heuristics on class numbers in hyperelliptic function fields. Math. Comp. 71(2002), no. 238, 837–861. (electronic) doi:10.1090/S0025-5718-01-01385-0Google Scholar
[27] Stein, A. and Teske, E., The parallelized Pollard kangaroo method in real quadratic function fields. Math. Comp. 71(2002), no. 238, 793–814. (electronic) doi:10.1090/S0025-5718-01-01343-6Google Scholar
[28] Stein, A. and Teske, E., Optimized baby-step giant-step methods. J. Ramanujan Math. Soc. 20(2005), no. 1, 1–32.Google Scholar
[29] Stein, A. and H. C.Williams, Some methods for evaluating the regulator of a real quadratic function field. Experiment. Math. 8(1999), no. 2, 119–133.Google Scholar
[30] Stichtenoth, H., Algebraic Function Fields and Codes. Springer-Verlag, Berlin, 1993.Google Scholar
[31] Tornheim, L., Minimal basis and inessential discriminant divisors for a cubic field. Pacific J. Math. 5(1955), 623–631.Google Scholar
[32] von Żylinśki, E., Zur Theorie der außerwesentlichen Diskriminantenteiler algebraischer Körper. Math. Ann. 73(1913), no. 2, 273–274. doi:10.1007/BF01456716Google Scholar
[33] Volcheck, E., Computing in the Jacobian of a plane algebraic curve. In: Algorithmic Number Theory. Lecture Notes in Comput. Sci. 877. Springer, Berlin, 1994, pp. 221–233.Google Scholar
[34] Weil, A., Sur les courbes algébraiques et les variétés qui s’en déduisent. Hermann, Paris, 1948.Google Scholar
[35] Weil, A., Variétés abéliennes et courbes algébraiques. Hermann, Paris, 1948.Google Scholar
[36] Wu, Q. and Scheidler, R., An explicit treatment of biquadratic function fields. Contrib. Discrete Math. 2(2007), no. 1, 43–60. (electronic)Google Scholar