P-Adic Hurwitz groups

Abstract

Herrlich showed that a Mumford curve of genus g>1 over the p-adic complex field Cp has at most 48(g−1), 24(g−1), 30(g−1) or 12(g−1) automorphisms as p=2,3,5 or p>5. The Mumford curves attaining these bounds are uniformised by normal subgroups of finite index in certain p-adic triangle groups Δp for p⩽5, or in a p-adic quadrangle group □p for p>5. The finite groups attaining these bounds are p-adic analogues of the Hurwitz groups arising from curves over C. We construct explicit infinite families of such groups as quotients of Δp and □p. These include groups of type PSL2(q) and PGL2(q), all arising as congruence quotients when p⩽5, and also various alternating and symmetric groups arising as noncongruence quotients of the groups Δp.