Let be a maximal -order in a division quaternion algebra over which is split at the place . The present article gives an algorithm to compute a fundamental domain for the action of the group of units on the Bruhat-Tits tree associated to . This action is a function field analog of the action of a co-compact Fuchsian group on the upper half plane. The algorithm also yields an explicit presentation of the group in terms of generators and relations. Moreover we determine an upper bound for its running time using that is almost Ramanujan.
Soit un -ordre maximal d’un corps de quaternions sur non-ramifié à la place . Cet article donne un algorithme pour calculer un domaine fondamental de l’action du groupe des unités sur l’arbre de Bruhat-Tits associé à , l’action étant un analogue en corps de fonctions de l’action d’un groupe cocompact Fuchsian sur le demi-plan supérieur. L’algorithme donne également une présentation explicite du groupe par générateurs et relations. En outre nous trouvons une borne supérieure pour le temps de calcul en utilisant que le graphe quotient est presque de Ramanujan.
@article{JTNB_2012__24_1_73_0, author = {B\"ockle, Gebhard and Butenuth, Ralf}, title = {On computing quaternion quotient graphs for function fields}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {73--99}, publisher = {Soci\'et\'e Arithm\'etique de Bordeaux}, volume = {24}, number = {1}, year = {2012}, doi = {10.5802/jtnb.789}, mrnumber = {2914902}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/jtnb.789/} }
TY - JOUR AU - Böckle, Gebhard AU - Butenuth, Ralf TI - On computing quaternion quotient graphs for function fields JO - Journal de théorie des nombres de Bordeaux PY - 2012 SP - 73 EP - 99 VL - 24 IS - 1 PB - Société Arithmétique de Bordeaux UR - http://archive.numdam.org/articles/10.5802/jtnb.789/ DO - 10.5802/jtnb.789 LA - en ID - JTNB_2012__24_1_73_0 ER -
%0 Journal Article %A Böckle, Gebhard %A Butenuth, Ralf %T On computing quaternion quotient graphs for function fields %J Journal de théorie des nombres de Bordeaux %D 2012 %P 73-99 %V 24 %N 1 %I Société Arithmétique de Bordeaux %U http://archive.numdam.org/articles/10.5802/jtnb.789/ %R 10.5802/jtnb.789 %G en %F JTNB_2012__24_1_73_0
Böckle, Gebhard; Butenuth, Ralf. On computing quaternion quotient graphs for function fields. Journal de théorie des nombres de Bordeaux, Volume 24 (2012) no. 1, pp. 73-99. doi : 10.5802/jtnb.789. http://archive.numdam.org/articles/10.5802/jtnb.789/
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