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Fundamental domains of some arithmetic groups over function fields. (English) Zbl 0858.11025
Let \({\mathbb{F}}_q\) be a finite field, \(A={\mathbb{F}}_q[T]\), \(K= \text{Quot} (A)\), \(K_\infty\) be the completion of \(K\) at the place \(\infty=1/T\). Let \(\Delta \subset GL_2(A)\) be one of the congruence subgroups and, for \({\mathcal N}\) a nonconstant element of \(A\), let \(\Gamma ({\mathcal N})\), \(\Gamma_1 ({\mathcal N})\), \(\Gamma_1^* ({\mathcal N})\), \(\Gamma_0 ({\mathcal N})\) be the subgroups of matrices congruent \(\text{mod } {\mathcal N}\) to \(\left( \begin{smallmatrix} 1 & 0 \\ 0 & 1 \end{smallmatrix} \right)\), \(\left( \begin{smallmatrix} 1 & * \\ 0 & 1 \end{smallmatrix} \right)\), \(\left( \begin{smallmatrix} 1 & * \\ 0 & * \end{smallmatrix} \right)\), \(\left( \begin{smallmatrix} * & * \\ 0 & * \end{smallmatrix} \right)\) respectively. Let \(\tau\) be the Bruhat-Tits tree of \(GL_2(K_\infty)\). One knows [following J. P. Serre, Trees (1980; Zbl 0548.20018)] that many arithmetic and geometric properties of the Drinfeld modular curve \(M_\Delta\) associated to \(\Delta\) are carried by the quotient graph \(\Delta \backslash \tau\). In this paper, the authors give the answers to natural questions about the graph \(\Delta \backslash \tau\), then about the curve \(M_\Delta\): general assertions about the structure of \(\Delta \backslash \tau\), procedure of calculation of this graph, formulae for numerical invariants as the number of cusps of \(M_\Delta\) or the genus of its completion... All details are given for \(\Delta = \Gamma_0 ({\mathcal N})\), the proofs for the other groups being almost the same and easier.

11F06 Structure of modular groups and generalizations; arithmetic groups
20H05 Unimodular groups, congruence subgroups (group-theoretic aspects)
11G20 Curves over finite and local fields
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