Hostname: page-component-7dd5485656-wxk4p Total loading time: 0 Render date: 2025-10-29T02:17:28.824Z Has data issue: false hasContentIssue false
  • English
  • Français

On lower bounds for the Ihara constants $A(2)$ and $A(3)$

Part of: Arithmetic algebraic geometry Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  03 May 2013

Iwan Duursma  and
Kit-Ho Mak
Iwan Duursma
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, 273 Altgeld Hall, MC-382, 1409 W. Green Street, Urbana, Illinois 61801, USA email duursma@math.uiuc.edu
Kit-Ho Mak
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, 273 Altgeld Hall, MC-382, 1409 W. Green Street, Urbana, Illinois 61801, USA email mak4@illinois.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Let $ \mathcal{X} $ be a curve over ${ \mathbb{F} }_{q} $ and let $N( \mathcal{X} )$, $g( \mathcal{X} )$ be its number of rational points and genus respectively. The Ihara constant $A(q)$ is defined by $A(q)= {\mathrm{lim~sup} }_{g( \mathcal{X} )\rightarrow \infty } N( \mathcal{X} )/ g( \mathcal{X} )$. In this paper, we employ a variant of Serre’s class field tower method to obtain an improvement of the best known lower bounds on $A(2)$ and $A(3)$.

Keywords

curves with many pointsasymptotic boundIhara constant

MSC classification

Secondary: 11G20: Curves over finite and local fields 14G15: Finite ground fields 14G05: Rational points

Information

Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 7 , July 2013 , pp. 1108 - 1128
Copyright
© The Author(s) 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Angles, B. and Maire, C., A note on tamely ramified towers of global function fields, Finite Fields Appl. 8 (2002), 207–215.CrossRefGoogle Scholar
Auer, R., Ray class fields of global function fields with many rational places, Acta Arith. 95 (2000), 97–122.Google Scholar
Cassels, J. W. S. and Fröhlich, A., Algebraic number theory (London Mathematical Society, Providence, RI, 2010).Google Scholar
Drinfel’d, V. G. and Vlăduţ, S. G., The number of points of an algebraic curve, Funktsional. Anal. i Prilozhen. 17 (1983), 68–69.Google Scholar
Garcia, A., Stichtenoth, H., Bassa, A. and Beelen, P., Towers of function fields over non-prime finite fields, Preprint, math.AG/1202.5922.Google Scholar
Gaschütz, W. and Newman, M. F., On presentations of finite $p$-groups, J. Reine Angew. Math. 245 (1970), 172–176.Google Scholar
Golod, E. S. and Safarevich, I. R., On the class field tower, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 261–272.Google Scholar
Hajir, F. and Maire, C., Asymptotically good towers of global fields, in European congress of mathematics, Barcelona, 2000, vol. II, Progress in Mathematics, vol. 202 (Birkhäuser, Basel, 2001), 207–218.Google Scholar
Hajir, F. and Maire, C., Extensions of number fields with wild ramification of bounded depth, Int. Math. Res. Not. IMRN 2002 (2002), 667–696.Google Scholar
Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, vol. 52 (Springer, New York, 1977).Google Scholar
Hoechsmann, K., Zum Einbettungsproblem, J. Reine Angew. Math. 229 (1968), 81–106.Google Scholar
Ihara, Y., Some remarks on the number of rational points of algebraic curves over finite fields, J. Fac. Sci. Univ. Tokyo Sect. IA 28 (1981), 721–724.Google Scholar
Kuhnt, T., Generalizations of Golod–Shavarevich and applications, PhD thesis, University of Illinois at Urbana-Champaign (2002).Google Scholar
Lang, S., Algebra, Graduate Texts in Mathematics, vol. 211, third edition (Springer, New York, 2002).CrossRefGoogle Scholar
Li, W.-C. W., Upper and lower bounds for $A(q)$, in Recent trends in coding theory and its applications, AMS/IP Studies in Advanced Mathematics, vol. 41 (American Mathematical Society, Providence, RI, 2007), 15–23.Google Scholar
Li, W.-C. W. and Maharaj, H., Coverings of curves with asymptotically many rational points, J. Number Theory 96 (2002), 232–256.Google Scholar
Neukirch, J., Über das Einbettungsproblem der algebraischen Zahlentheorie, Invent. Math. 21 (1973), 59–116.CrossRefGoogle Scholar
Neukirch, J., Schmidt, A. and Wingberg, K., Cohomology of number fields, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 323, second edition (Springer, Berlin, 2008).Google Scholar
Niederreiter, H. and Xing, C., Towers of global function fields with asymptotically many rational places and an improvement on the Gilbert–Varshamov bound, Math. Nachr. 195 (1998), 171–186.Google Scholar
Niederreiter, H. and Xing, C., Curve sequences with asymptotically many rational points, in Applications of curves over finite fields, Seattle, WA, 1997, Contemporary Mathematics, vol. 245 (American Mathematical Society, Providence, RI, 1999), 3–14.Google Scholar
Niederreiter, H. and Xing, C., Rational points on curves over finite fields: theory and applications, London Mathematical Society Lecture Note Series, vol. 285 (Cambridge University Press, Cambridge, 2001).Google Scholar
Schoof, R., Algebraic curves over ${\mathbf{F} }_{2} $ with many rational points, J. Number Theory 41 (1992), 6–14.CrossRefGoogle Scholar
Serre, J.-P., Local fields, Graduate Texts in Mathematics, vol. 67 (Springer, New York, 1979).Google Scholar
Serre, J.-P., Rational points on curves over finite fields, lecture notes, Harvard University (1985).Google Scholar
Serre, J.-P., Galois cohomology, Springer Monographs in Mathematics, English edition (Springer, Berlin, 2002).Google Scholar
Stichtenoth, H., Algebraic function fields and codes, Graduate Texts in Mathematics, vol. 254, second edition (Springer, Berlin, 2009).Google Scholar
Temkine, A., Hilbert class field towers of function fields over finite fields and lower bounds for $A(q)$, J. Number Theory 87 (2001), 189–210.Google Scholar
Tsfasman, M. A., Vlăduţ, S. G. and Zink, Th., Modular curves, Shimura curves, and Goppa codes, better than Varshamov–Gilbert bound, Math. Nachr. 109 (1982), 21–28.Google Scholar
Weil, A., Sur les courbes algébriques et les variétés qui s’en déduisent, Actualités Sci. Ind., no. 1041; Publ. Inst. Math. Univ. Strasbourg, vol. 7 (1945) (Hermann, Paris, 1948).Google Scholar
Weibel, C. A., An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38 (Cambridge University Press, Cambridge, 1994).Google Scholar
Xing, C. and Yeo, S. L., Algebraic curves with many points over the binary field, J. Algebra 311 (2007), 775–780.Google Scholar