Constructing smooth and fully faithful tropicalizations for Mumford curves

The tropicalization of an algebraic variety X is a combinatorial shadow of X, which is sensitive to a closed embedding of X into a toric variety. Given a good embedding, the tropicalization can provide a lot of information about X. We construct two types of these good embeddings for Mumford curves: fully faithful tropicalizations, which are embeddings such that the tropicalization admits a continuous section to the associated Berkovich space $$X^{{{\,\mathrm{an}\,}}}$$ X an of X, and smooth tropicalizations. We also show that a smooth curve that admits a smooth tropicalization is necessarily a Mumford curve. Our key tool is a variant of a lifting theorem for rational functions on metric graphs.

MULTILOOP CALCULUS IN P-ADIC STRING THEORY AND BRUHAT-TITS TREES

We treat the open p-adic string world sheet as a coset space F=T/Γ, where T is the Bruhat-Tits tree for the p-adic linear group GL (2, ℚp) and Γ⊂ PGL (2, ℚp) is some Schottky group. 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 tree T. We find the amplitudes for emission processes of the tachyon states from the boundary.