P-adic Schwarzian triangle groups of Mumford type

Abstract

In this article we discuss a certain p-adic analogue of classical Schwarzian triangle groups, an analogue which is related to Mumford's uniformization of p-adic analytic curves. p-adic Schwarzian triangle groups are defined to be the Galois groups of analytic coverings over projective line with precisely 3 branch points. We say that a p-adic triangle group is of Mumford type if the corresponding universal covering is given by a certain locally compact analytic subspace in projective line, related to Mumford's uniformization. The main theorem provides a complete classification of p-adic triangle groups of Mumford type; our list of these groups contains some of the arithmetic p-adic triangle groups which are discussed by Yves Andre. Notably, we deduce that p-adic triangle groups of Mumford type exist only if p=2,3,5.