Quasi-inner automorphisms of Drinfeld modular groups (Q6594729)

Let \(K\) be an algebraic function field of one variable with constant field \(\mathbb{F}_{q}\), the finite field of order \(q\), \(\infty\) a fixed place of \(K\), and \(\delta\) its degree. The ring \(A\) of all those elements of \(K\) which are integral outside \(\infty\) is a Dedekind domain. Denote by \(K_{\infty}\) the completion of \(K\) with respect to \(\infty\), and let \(C_{\infty}\) be the \(\infty\)-completion of an algebraic closure of \(K_{\infty}\). The group \(\mathrm{GL}_{2}(K_{\infty})\) (and its subgroup \(G=\mathrm{GL}_{2}(A)\)) acts as Möbius transformations on \(C_{\infty}\), \(K_{\infty}\) and hence \(\Omega= C_{\infty} \setminus K_{\infty}\), the Drinfeld upper halfplane. This is part of a far-reaching analogy, initiated by \textit{V. G. Drinfeld}, in [Math. USSR, Sb. 23(1974), 561--592 (1976; Zbl 0321.14014)], where \(\mathbb{Q}\), \(\mathbb{R}\), \(\mathbb{C}\) are replaced by \(K\), \(K_{\infty}\), \(C_{\infty}\), respectively. The roles of the classical upper half plane (in \(\mathbb{C}\)) and the classical modular group \(\mathrm{SL}_{2}(\mathbb{Z})\) are assumed by \(\Omega\) and \(G\), respectively.\N\NThe normalizer of \(G\) in \(\mathrm{GL}_{2}(K)\) gives rise to automorphisms of \(G\), which the authors refer to as quasi-inner. Modulo the inner automorphisms of \(G\), they form a group \(\mathrm{Quinn}(G)\) which is isomorphic to \(\mathrm{Cl}(A)_{2}\), the 2-torsion in the ideal class group \(\mathrm{Cl}(A)\). The group \(\mathrm{Quinn}(G)\) acts on all kinds of objects associated with \(G\) (for example, it acts freely on the cusps and elliptic points of \(G\)). If \(\mathcal{T}\) is the associated Bruhat-Tits tree, the elements of \(\mathrm{Quinn}(G)\) induce non-trivial automorphisms of the quotient graph \(G \setminus \mathcal{T}\), generalizing an earlier result of \textit{J. P. Serre} (see [Trees. Transl. from the French by John Stillwell. Berlin-Heidelberg-New York: Springer-Verlag (1980; Zbl 0548.20018)]).\N\NThe main result of this paper is Theorem 1.1: The group \(\mathrm{Quinn}(G)\) acts freely on (i) \(\mathrm{Cusp}(G)\), (ii) \(\mathrm{Ell}(G)\) if \(\delta\) is odd.\N\NThe group \(\mathrm{Quinn}(G)\) can be embedded as a subgroup \(\mathrm{Ell}(G)^{=}\) (resp. \(\mathrm{Cl}(A)_{2}\) of \(\mathrm{Ell}(G)\) (resp. of \(\mathrm{Cusp}(G)\)). The authors show that the action of \(\mathrm{Quinn}(G)\) is equivalent to multiplication by the elements of the subgroup. Hence they prove Corollary 1.2: The group \(\mathrm{Quinn}(G)\) acts freely and transitively on (i) \(\mathrm{Cl}(A)_{2}\), (ii) \(\mathrm{Ell}(G)^{=}\) if \(\delta\) is odd.\N\NThey also prove Theorem 1.4: Every non-trivial element of \(\mathrm{Quinn}(G)\) determines an automorphism of \(G \setminus \mathcal{T}\) of order 2 which preserves the structure of all its vertex and edge stabilizers. And Theorem 1.6: (a) The group \(\mathrm{Quinn}(G)\) acts freely and transitively on \(\{ \widetilde{v}\in \mathrm{vert}(G\setminus \mathcal{T}) \mid G_{v} \simeq \mathrm{GL}_{2}(\mathbb{F}_{q}) \}\). (b) Suppose that \(\delta\) is odd and that \(\ker \overline{N}\) has no element of order 4. Then \(\mathrm{Quinn}(G)\) acts freely on \(\{ \widetilde{v}\in \mathrm{vert}(G\setminus \mathcal{T}) \mid G_{v} \simeq \mathbb{F}_{q^{2}}^{\ast} \}\).