On computing quaternion quotient graphs for function fields
Journal de théorie des nombres de Bordeaux, Tome 24 (2012) no. 1, pp. 73-99.

Soit Λ un 𝔽 q [T]-ordre maximal d’un corps de quaternions sur 𝔽 q (T) non-ramifié à la place . 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 . 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.

DOI : 10.5802/jtnb.789
Gebhard Böckle  ; Ralf Butenuth 1

1 Interdisziplinäres Zentrum für wissenschaftliches Rechnen Im Neuenheimer Feld 368 69120 Heidelberg, Germany
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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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