I am not an algebraic geometer and I know next to nothing about moduli spaces, so the following question might be trivial, or well covered in the literature, only I didn't recognize it. So I am basically asking to be pointed to the right literature.

Let $K$ be a local field of positive characteristic. In his celebrated Compositio paper of 1972, David Mumford shows that for a torsion-free discrete subgroup $\Gamma$ of $PGL_2(K)$ there exists a curve $C$ and an analytic isomorphism of $\Omega/\Gamma$ with the $K$-points of that curve, where $\Omega$ is the open complement of the limit point set of $\Gamma$ in ${\mathbb P}_1(K)$.

There are, however, also discrete, even arithmetic subgroups $\Gamma$ of $PGL_2(K)$, which are far from torsion-free, like for example $\Gamma=PGL_2(A)$, where, say $K={\mathbb F}((t))$ and $A={\mathbb F}[1/t]$. Is there a canonical algebraic structure on $\Omega/\Gamma$ in this case as well? Is there an algebro-geometric interpretation, as in Mumford's case, of the quotient $B/\Gamma$, where $B$ is the Bruhat-Tits tree of $PGL_2(K)$?

  • $\begingroup$ Something is odd here (lack of finite generation assumption). However, maybe an infinite degree cover of an algebraic curve can be still algebraic. $\endgroup$ – Misha Jun 19 '14 at 3:27

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