Green’s functions on Mumford curves

We study an analogue of a Green’s function on Mumford curves. The meromorphic continuation of this Green’s function comes from a “differential equation,” and we interpret the special value at \(s=0\) as a “volume” of the corresponding curve. Using harmonic analysis on Bruhat-Tits trees, we derive a “Kronecker limit formula” for the Green’s functions in question, which connects the derivative at \(s=0\) with the Manin–Drinfeld theta functions. Following Gross’ description of Neron’s local height pairing on Mumford curves, the special derivative here then equals twice of the Neron’s local height with sign changed.