Multiloop calculations in p-adic string theory and Bruhat-Tits trees. 1

The open p-adic string world sheet as a coset space F=T/Γ, where T is the Bruhat-Tits three for the p-adic linear group GL(2.Qp) and Γ is contained it PGL(2.Qp) is some Schottky group is treated. The boundary of this world sheet corresponds to p-adic Mumford curve of finite genus. The string dynamics is governed by the local gaussian action on the coset space F. The tachyon amplitudes expressed in terms of p-adic Θ-functions are proposed for the Mumford curve of arbitrary genus and compared them with the corresponding usual archimedian amplitudes. 25 refs.; 5 figs