On computing quaternion quotient graphs for function fields

Gebhard Böckle; Ralf Butenuth

doi:10.5802/jtnb.789

Gebhard Böckle ; Ralf Butenuth

Journal de Théorie des Nombres de Bordeaux, Tome 24 (2012) no. 1, pp. 73-99.

Résumé
Abstract

Soit $Λ$ un $𝔽_{q} [T]$ -ordre maximal d’un corps de quaternions sur $𝔽_{q} (T)$ non-ramifié à la place $\infty$ . Cet article donne un algorithme pour calculer un domaine fondamental de l’action du groupe des unités $Λ^{*}$ sur l’arbre de Bruhat-Tits $𝒯$ associé à ${PGL}_{2} (𝔽_{q} ((1 / T)))$ , 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 $Λ^{*}$ par générateurs et relations. En outre nous trouvons une borne supérieure pour le temps de calcul en utilisant que le graphe quotient $Λ^{*} ∖ 𝒯$ est presque de Ramanujan.

Let $Λ$ be a maximal $𝔽_{q} [T]$ -order in a division quaternion algebra over $𝔽_{q} (T)$ which is split at the place $\infty$ . The present article gives an algorithm to compute a fundamental domain for the action of the group of units $Λ^{*}$ on the Bruhat-Tits tree $𝒯$ associated to ${PGL}_{2} (𝔽_{q} ((1 / T)))$ . 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 $Λ^{*}$ in terms of generators and relations. Moreover we determine an upper bound for its running time using that $Λ^{*} ∖ 𝒯$ is almost Ramanujan.

Reçu le : 2010-10-21
Publié le : 2012-04-12

MR 2914902 | Zbl pre06075023

DOI : https://doi.org/10.5802/jtnb.789

@article{JTNB_2012__24_1_73_0,
     author = {Gebhard B\"ockle and Ralf Butenuth},
     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},
     zbl = {pre06075023},
     mrnumber = {2914902},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.789/}
}

Gebhard Böckle; Ralf Butenuth. On computing quaternion quotient graphs for function fields. Journal de Théorie des Nombres de Bordeaux, Tome 24 (2012) no. 1, pp. 73-99. doi : 10.5802/jtnb.789. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.789/

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