Green's functions on Mumford curves
The paper in question introduces and studies an analogue of Green's function on Mumford curves. Let us introduce some notation. \(F\) denotes a non-Archimedean local field with finite residue field \(\mathbb{F}_q\) and completed algebraic closure \(\mathbb{C}_F\). Put \(\mathcal{H}:= \mathbb{C}_F\setminus F\), the Drinfeld upper half-plane, and let \(\Gamma\) be a discrete subgroup of \[ \mathrm{PGL}_2^+(F) = \{g \in \mathrm{PGL}_2(F)\mid \det(g) \text{ has even order}\}. \] Let further \(\lambda: \mathcal{H} \longrightarrow\mathcal{T}(\mathbb{Q})\) be the building map onto the points of the Bruhat-Tits building \(\mathcal{T}\) with rational barycentric coordinates. We suppose that \(\Gamma\setminus \mathcal{T}\) is a finite graph (that is, \(\Gamma\) is cocompact in \(\mathrm{PGL}_2(F)\)). For \(z \in \mathcal{H}\) let \(|z|_i := \inf_{a\in F}|z-a|\) be the ``imaginary part'', and for \(z,w \in \mathcal{H}\), \[ d(z,w):= |z-w|^2/|z|_i|w_i| \] the ``hyperbolic distance''. It satisfies \[ \max\{ \log_q d(z,w),0\} = \ell(\lambda(z),\lambda(w)) \] with the distance function \(\ell\) on \(\mathcal{T}(\mathbb{Q})\). The Schottky group \(\Gamma\) determines a Mumford curve \(X_{\Gamma}\) over \(\mathbb{C}_F\) with points \(X_{\Gamma}(\mathbb{C}_F) = \Gamma \setminus \mathcal{H}\). The author defines the Green's function \[ G_{\Gamma}(z,w,s) = \sum_{\gamma \in \Gamma} d(z,\gamma w)^{-s}, \] where \(z,w \in \mathcal{H}\), \(\Gamma z \not= \Gamma w\), and \(s \in \mathbb{C}\). This is (modulo convergence) a symmetric function on \(X_{\Gamma}(\mathbb{C}_F) \times X_{\Gamma}(\mathbb{C}_F)\) minus the diagonal. The main result is Theorem 1.1, which states: {\parindent=8mm \begin{itemize}\item[(i)] \(G_{\Gamma}(z,w,s)\) converges absolutely for \(\mathrm{Re}(s) >1\) and has a meromorphic extension to \(\mathbb{C}\); \item[(ii)] it is holomorphic at \(s=0\) and may be explicitly evaluated there with a value independent of \(z,w\); \item[(iii)] given divisors \(D,E\) of degree 0 on \(X_{\Gamma}\) with disjoint support, \(G_{\Gamma}(D,E,s)\) may be defined through linear extension; then \[ \frac{\partial}{\partial s} G_{\Gamma}(D,E,s)|_{s=0} = -2 \langle D,E \rangle \] holds, where \(\langle .\,,. \rangle\) is the local height pairing. \end{itemize}} Also, a variant of Green's function on \(\mathcal{T}(\mathbb{Q})\) is given. These are related in Theorem 1.2 and 1.3 with Manin-Drinfeld theta functions on \(\mathcal{H}\), and with certain functions stemming from the geometry of \(\mathcal{T}\).
scientific article; zbMATH DE number 6856715
scientific article