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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.

##### MSC:
 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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