On computing quaternion quotient graphs for function fields

Gebhard Böckle; Ralf Butenuth[1]

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

Journal de Théorie des Nombres de Bordeaux (2012)

  • Volume: 24, Issue: 1, page 73-99
  • ISSN: 1246-7405

Abstract

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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.

How to cite

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Böckle, Gebhard, and Butenuth, Ralf. "On computing quaternion quotient graphs for function fields." Journal de Théorie des Nombres de Bordeaux 24.1 (2012): 73-99. <http://eudml.org/doc/251039>.

@article{Böckle2012,
abstract = {Let $\Lambda $ be a maximal $\{\{\mathbb\{F\}\}\,\!\{\}_q\}[T]$-order in a division quaternion algebra over $\{\{\mathbb\{F\}\}\,\!\{\}_q\}(T)$ 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 $\Lambda ^*$ on the Bruhat-Tits tree $\mathcal\{T\}$ associated to $\{\rm PGL\}_2(\{\{\mathbb\{F\}\}\,\!\{\}_q\}((1/T)))$. 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 $\Lambda ^*$ in terms of generators and relations. Moreover we determine an upper bound for its running time using that $\Lambda ^*\backslash \mathcal\{T\}$ is almost Ramanujan.},
affiliation = {Interdisziplinäres Zentrum für wissenschaftliches Rechnen Im Neuenheimer Feld 368 69120 Heidelberg, Germany},
author = {Böckle, Gebhard, Butenuth, Ralf},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
month = {3},
number = {1},
pages = {73-99},
publisher = {Société Arithmétique de Bordeaux},
title = {On computing quaternion quotient graphs for function fields},
url = {http://eudml.org/doc/251039},
volume = {24},
year = {2012},
}

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
DA - 2012/3//
PB - Société Arithmétique de Bordeaux
VL - 24
IS - 1
SP - 73
EP - 99
AB - Let $\Lambda $ be a maximal ${{\mathbb{F}}\,\!{}_q}[T]$-order in a division quaternion algebra over ${{\mathbb{F}}\,\!{}_q}(T)$ 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 $\Lambda ^*$ on the Bruhat-Tits tree $\mathcal{T}$ associated to ${\rm PGL}_2({{\mathbb{F}}\,\!{}_q}((1/T)))$. 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 $\Lambda ^*$ in terms of generators and relations. Moreover we determine an upper bound for its running time using that $\Lambda ^*\backslash \mathcal{T}$ is almost Ramanujan.
LA - eng
UR - http://eudml.org/doc/251039
ER -

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