×

zbMATH — the first resource for mathematics

On generators of arithmetic groups over function fields. (English) Zbl 1231.11047
Let \({\mathbb F}_q\) be the finite field with \(q\) elements, \(A={\mathbb F}_q[T]\) the polynomial ring, and \(F={\mathbb F}_q(T)\) the field of rational functions in an indeterminate \(T\). Let further \(D\) be a quaternion division algebra over \(k\) split at the place at infinity, \(\Lambda\) an \(A\)-order in \(D\), and \(\Gamma = \Lambda^{\ast}\) its multiplicative group.
Suppose that \(q\) is odd (it is known that quaternion algebras behave differently in characteristic 2), and let \({\mathfrak r} \in A\) be the discriminant of \(D\) (the product of the monic generators the primes in \(A\) where \(D\) ramifies). Then there exists \({\mathfrak a} \in A\) such that \(D\) is the \(F\)-algebra with basis vectors \(1,i,j,ij\) that satisfy the relations \[ i^2 = {\mathfrak a},\;j^2 = {\mathfrak r},\;ij = -ji. \] Now let \(\Lambda\) be the \(A\)-order \(\Lambda = A \oplus Ai\oplus Aj \oplus Aij\). Then \[ \Gamma = \{a+bi+cj+dij\mid a,b,c,d \in A;\;a^2-{\mathfrak a}b^2-{\mathfrak r}c^2+{\mathfrak a}{\mathfrak r}d^2 \in {\mathbb F}_q^{\ast}\}. \] In his main result Theorem 1.2, the author shows:
The finite set \[ S=\{\gamma \in \Gamma~|~\|\gamma\| \leq 4 \deg ({\mathfrak a}{\mathfrak r}) +6\} \] generates \(\Gamma\), where \(\|a+bi+cj+dij\|\) is the maximum of the degrees of \(a,b,c,d\).
The author further discusses related problems:
1.
How large is the set \(S\)?
2.
To what extent can the set \(S\) be reduced to a generating subset \(S'\) with a description still essentially independent of \(\Gamma\), i.e., with absolute constants \(\sigma,\delta\) such that \[ S' = \{\gamma \in \Gamma~|~\|\gamma\| \leq \sigma~\deg ({\mathfrak a}{\mathfrak r}) + \delta\}? \]
The proof of Theorem 1.2 uses the action of \(\Gamma\) on the attached Bruhat-Tits tree and a quantification of the discontinuity of this action.

MSC:
11F06 Structure of modular groups and generalizations; arithmetic groups
11G18 Arithmetic aspects of modular and Shimura varieties
20E08 Groups acting on trees
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Alsina M., Quaternion Orders, Quadratic Forms and Shimura Curves (2004)
[2] Chalk J., Proc. Roy. Soc. Edinburgh Sect. A 72 pp 317–
[3] DOI: 10.1007/BF02568000 · Zbl 0884.11025
[4] DOI: 10.1090/S0025-5718-99-01167-9 · Zbl 0937.11016
[5] DOI: 10.1007/978-3-0346-0332-4 · Zbl 1183.22001
[6] DOI: 10.1007/BF02772543 · Zbl 1087.05036
[7] DOI: 10.1007/3-540-29593-3 · Zbl 1159.11014
[8] DOI: 10.1007/978-1-4757-6046-0
[9] DOI: 10.1007/978-1-4757-5673-9
[10] Serre J.-P., Springer Monographs in Mathematics, in: Trees (2003)
[11] Vignéras M.-F., Lecture Notes in Mathematics 800, in: Arithmétiques des Algèbres de Quaternions (1980) · Zbl 0422.12008
[12] Voight J., J. Théor. Nombres Bordeaux 21 pp 469–
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.