Nous donnons des bornes inférieures sur le nombre de diviseurs effectifs de degré par rapport au nombre de places d’un certain degré d’un corps de fonctions algébriques de genre défini sur un corps fini. Nous déduisons des bornes inférieures du nombre de classes qui améliorent les bornes de Lachaud-Martin-Deschamps et des bornes inférieures asymptotiques atteignant celles de Tsfasman-Vladut. Nous donnons des exemples de tours de corps de fonctions algébriques ayant un grand nombre de classes.
We give lower bounds on the number of effective divisors of degree with respect to the number of places of certain degrees of an algebraic function field of genus 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.
Mots clés : Finite field, function field, class number
@article{JTNB_2012__24_3_505_0,
author = {Ballet, St\'ephane and Rolland, Robert},
title = {Lower bounds on the class number of algebraic function fields defined over any finite field},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {505--540},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {24},
number = {3},
year = {2012},
doi = {10.5802/jtnb.809},
zbl = {1273.14052},
mrnumber = {3010627},
language = {en},
url = {http://www.numdam.org/articles/10.5802/jtnb.809/}
}
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 PY - 2012 SP - 505 EP - 540 VL - 24 IS - 3 PB - Société Arithmétique de Bordeaux UR - http://www.numdam.org/articles/10.5802/jtnb.809/ DO - 10.5802/jtnb.809 LA - en ID - JTNB_2012__24_3_505_0 ER -
%0 Journal Article %A Ballet, Stéphane %A Rolland, Robert %T Lower bounds on the class number of algebraic function fields defined over any finite field %J Journal de théorie des nombres de Bordeaux %D 2012 %P 505-540 %V 24 %N 3 %I Société Arithmétique de Bordeaux %U http://www.numdam.org/articles/10.5802/jtnb.809/ %R 10.5802/jtnb.809 %G en %F JTNB_2012__24_3_505_0
Ballet, Stéphane; Rolland, Robert. Lower bounds on the class number of algebraic function fields defined over any finite field. Journal de théorie des nombres de Bordeaux, Tome 24 (2012) no. 3, pp. 505-540. doi : 10.5802/jtnb.809. http://www.numdam.org/articles/10.5802/jtnb.809/
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