On computing quaternion quotient graphs for function fields

Böckle, Gebhard; Butenuth, Ralf

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

Soit $\Lambda$ un $𝔽{}_q\left[T\right]$-ordre maximal d’un corps de quaternions sur $𝔽{}_q\left(T\right)$ non-ramifié à la place $\infty$. Cet article donne un algorithme pour calculer un domaine fondamental de l’action du groupe des unités ${\Lambda }^{*}$ sur l’arbre de Bruhat-Tits $𝒯$ associé à ${\mathrm{PGL}}_2\left(𝔽{}_q\left((1/T)\right)\right)$, 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 ${\Lambda }^{*}$ par générateurs et relations. En outre nous trouvons une borne supérieure pour le temps de calcul en utilisant que le graphe quotient ${\Lambda }^{*}\setminus 𝒯$ est presque de Ramanujan.

Let $\Lambda$ be a maximal $𝔽{}_q\left[T\right]$-order in a division quaternion algebra over $𝔽{}_q\left(T\right)$ 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 ${\Lambda }^{*}$ on the Bruhat-Tits tree $𝒯$ associated to ${\mathrm{PGL}}_2\left(𝔽{}_q\left((1/T)\right)\right)$. 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 ${\Lambda }^{*}$ in terms of generators and relations. Moreover we determine an upper bound for its running time using that ${\Lambda }^{*}\setminus 𝒯$ is almost Ramanujan.

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