Quasi-inner automorphisms of Drinfeld modular groups

Abstract

Let $A$ be the set of elements in an algebraic function field $K$ over ${\mathbb{F}}_q$ which are integral outside a fixed place $\infty$. Let $G={\mathrm{GL}}_2(A)$ be a Drinfeld modular group. The normalizer of $G$ in ${\mathrm{GL}}_2(K)$, where $K$ is the quotient field of $A$, gives rise to automorphisms of $G$, which we 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 $G\backslash \mathcal{T}$, generalizing an earlier result of Serre. It is known that the ends of $G\backslash \mathcal{T}$ are in one-to-one correspondence with the cusps of $G$. Consequently, $\mathrm{Quinn}(G)$ acts freely on the ends. In addition, $\mathrm{Quinn}(G)$ acts transitively on those ends which are in one-to-one correspondence with the vertices of $G\backslash \mathcal{T}$ whose stabilizers are isomorphic to ${\mathrm{GL}}_2({\mathbb{F}}_q)$.