Lower bounds on the class number of algebraic function fields defined over any finite field

Stéphane Ballet[1]; Robert Rolland[1]

  • [1] Aix-Marseille Université Institut de Mathématiques de Luminy Équipe Arithmétique et Théorie de l’Information et Groupe d’Études et Recherche en Informatique des Systèmes Communicants Sécurisés. Case 907 F13288 Marseille cedex 9 France

Journal de Théorie des Nombres de Bordeaux (2012)

  • Volume: 24, Issue: 3, page 505-540
  • ISSN: 1246-7405

Abstract

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We give lower bounds on the number of effective divisors of degree ≤ g - 1 with respect to the number of places of certain degrees of an algebraic function field of genus g defined over a finite field. We deduce lower bounds for the class number which improve the Lachaud - Martin-Deschamps bounds and asymptotically reaches the Tsfasman-Vladut bounds. We give examples of towers of algebraic function fields having a large class number.

How to cite

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Ballet, Stéphane, and Rolland, Robert. "Lower bounds on the class number of algebraic function fields defined over any finite field." Journal de Théorie des Nombres de Bordeaux 24.3 (2012): 505-540. <http://eudml.org/doc/251069>.

@article{Ballet2012,
abstract = {We give lower bounds on the number of effective divisors of degree $\le g-1$ with respect to the number of places of certain degrees of an algebraic function field of genus $g$ defined over a finite field. We deduce lower bounds for the class number which improve the Lachaud - Martin-Deschamps bounds and asymptotically reaches the Tsfasman-Vladut bounds. We give examples of towers of algebraic function fields having a large class number.},
affiliation = {Aix-Marseille Université Institut de Mathématiques de Luminy Équipe Arithmétique et Théorie de l’Information et Groupe d’Études et Recherche en Informatique des Systèmes Communicants Sécurisés. Case 907 F13288 Marseille cedex 9 France; Aix-Marseille Université Institut de Mathématiques de Luminy Équipe Arithmétique et Théorie de l’Information et Groupe d’Études et Recherche en Informatique des Systèmes Communicants Sécurisés. Case 907 F13288 Marseille cedex 9 France},
author = {Ballet, Stéphane, Rolland, Robert},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Finite field; function field; class number; finite field},
language = {eng},
month = {11},
number = {3},
pages = {505-540},
publisher = {Société Arithmétique de Bordeaux},
title = {Lower bounds on the class number of algebraic function fields defined over any finite field},
url = {http://eudml.org/doc/251069},
volume = {24},
year = {2012},
}

TY - JOUR
AU - Ballet, Stéphane
AU - Rolland, Robert
TI - Lower bounds on the class number of algebraic function fields defined over any finite field
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2012/11//
PB - Société Arithmétique de Bordeaux
VL - 24
IS - 3
SP - 505
EP - 540
AB - We give lower bounds on the number of effective divisors of degree $\le g-1$ with respect to the number of places of certain degrees of an algebraic function field of genus $g$ defined over a finite field. We deduce lower bounds for the class number which improve the Lachaud - Martin-Deschamps bounds and asymptotically reaches the Tsfasman-Vladut bounds. We give examples of towers of algebraic function fields having a large class number.
LA - eng
KW - Finite field; function field; class number; finite field
UR - http://eudml.org/doc/251069
ER -

References

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  2. Stéphane Ballet and Robert Rolland, Minorations du nombre de classes des corps de fonctions algébriques définis sur un corps fini. Comptes Rendus de l’Académie des Sciences (Paris), Ser.I, 349 (2011), 709–712. Zbl1239.11131MR2825926
  3. Ignacio Cascudo, Ronald Cramer, Chaoping Xing, An Yang, Asymptotic bound for Multiplication complexity in the extensions of small finite fields. IEEE Transactions on Information Theory, to appear. 
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  11. Henning Stitchtenoth, Algebraic function fields and codes. Springer, Graduate texts in Mathematics, 254, second edition, 2009. Zbl1155.14022MR2464941
  12. Michael Tsfasman, Some remarks on the asymptotic number of points. In H. Stichtenoth and M.A. Tsfasman, editors, Coding Theory and Algebraic Geometry, volume 1518 of Lecture Notes in Mathematics, pages 178–192, Berlin, 1992. Springer-Verlag. Proceedings of AGCT-3 conference, June 17-21, 1991, Luminy. Zbl0745.00062MR1186424
  13. Michael Tsfasman and Serguei Vladut, Asymptotic properties of zeta-functions. Journal of Mathematical Sciences 84 (1997) (5), 1445–1467. Zbl0919.11045MR1465522
  14. Michael Tsfasman and Serguei Vladut, Infinite global fields and the generalized Brauer-Siegel theorem. Mosc. Math. J. 2 (2002) (2), 329–402. Zbl1004.11037MR1944510
  15. Michael Tsfasman, Serguei Vladut, and Dmitry Nogin, Algebraic Geometric Codes: Basic Notions, volume 139 of Mathematical Surveys and Monographs. American Mathematical Society, 2007. Zbl1127.94001MR2339649
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