A 𝑝-adic analogue of the Gauss-Bonnet theorem for certain Mumford curves

Abstract

If K K is a local field, L L a quadratic extension, Γ \Gamma a Schottky group co-compact in PG L 2 ( K ) {\text {PG}}{{\text {L}}_2}(K) then the quotient L − K / Γ L - K/\Gamma corresponds to the L L -points of a Mumford curve. In this paper we calculate ∫ L − K / Γ d M \int _{L - K/\Gamma } {d\mathcal {M}} where M \mathcal {M} is an PG L 2 ( K ) {\text {PG}}{{\text {L}}_2}(K) invariant measure on L − K L - K , in terms of the genus of the corresponding curve.