Multiloop calculations inp-adic string theory and Bruhat-Tits trees

We treat the openp-adic string world sheet as a coset spaceF=T/Γ, whereT is the Bruhat-Tits tree for thep-adic linear groupGL(2, ℚp) and Γ⊂PGL(2, ℚp) is some Schottky group. The boundary of this world sheet corresponds to ap-adic Mumford curve of finite genus. The string dynamics is governed by the local gaussian action on the coset spaceF. The tachyon amplitudes expressed in terms ofp-adic θ-functions are proposed for the Mumford curve of arbitrary genus. We compare them with the corresponding usual archimedean amplitudes. The sum over moduli space of the algebraic curves is conjectured to be expressed in the arithmetic surface terms. We also give the necessary mathematical background including the Mumford approach top-adic algebraic curves. The connection of the problem of closedp-adic strings with the considered topics is discussed.

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