Quasi-inner automorphisms of Drinfeld modular groups

TL;DR

This paper studies the automorphisms of Drinfeld modular groups arising from their normalizers, revealing their structure as 2-torsion in class groups and their actions on associated geometric objects.

Contribution

It introduces the concept of quasi-inner automorphisms of Drinfeld modular groups and characterizes their structure and actions on cusps, elliptic points, and quotient graphs.

Findings

Quasi-inner automorphisms form a group isomorphic to the 2-torsion of the class group.

These automorphisms act freely on cusps and elliptic points.

They induce non-trivial automorphisms of the quotient Bruhat-Tits tree.

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 $G=GL_2(A)$ be a {\it Drinfeld modular group}. The normalizer of $G$ in $GL_2(K)$, where $K$ is the quotient field of $A$, gives rise to automorphisms of $G$, which we refer to as {\it quasi-inner}. Modulo the inner automorphisms of $G$ they form a group $Quinn(G)$ which is isomorphic to ${\mathrm Cl}(A)_2$, the $2$-torsion in the ideal class group ${\mathrm Cl}(A)$. The group $Quinn(G)$ 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 $Quinn(G)$ induce non-trivial automorphisms of the quotient graph $G\setminus{\mathcal T}$, generalizing an earlier result of Serre. It is known that the ends of $G\setminus{\mathcal T}$ are in one-one correspondence with the cusps of $G$. Consequently $Quinn(G)$ acts freely on the ends. In addition $Quinn(G)$ acts transitively on those ends which are in one-one correspondence with the vertices of $G\setminus{\mathcal T}$ whose stabilizers are isomorphic to $GL_2({\mathbb F}_q)$.