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

Freije, Richard M.

doi:10.1090/S0002-9939-1989-0972230-7

A $p$-adic analogue of the Gauss-Bonnet theorem for certain Mumford curves
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by Richard M. Freije PDF

Proc. Amer. Math. Soc. 107 (1989), 323-332 Request permission

Abstract:

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

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