Green's functions for Drinfeld modular curves
Let \(\mathbb{F}_q\) be a finite field, \(K= \mathbb{F}_q(T)\) be a rational function field and denote by \(\infty\) the place \(1/T\) of \(K\). Let \(C_\infty\) be the completion of an algebraic closure of \(K_\infty= \mathbb{F}_q ((1/T))\). The Drinfeld upper half plane is \(\Omega= \mathbb{P}_{C_\infty}^1 (C_\infty)- \mathbb{P}_{C_\infty}^1 (K_\infty)\) and a Drinfeld modular curve \(M_\Gamma\) is the completion of a quotient \(\Gamma\setminus \Omega\), where \(\Gamma\subset PGL_2 (\mathbb{F}_q [T])\) is an arithmetic group. The author proves that Néron's local pairing of \(M_\Gamma\) can be described by an analytic function on \(\Omega\times \Omega- \Delta\) \((\Delta\) is the diagonal), which is a non-Archimedean Green's function, a notion first introduced by \textit{T. Chinburg} and \textit{R. Rumely} [J. Reine Angew. Math. 434, 1-44 (1993; Zbl 0756.14013)], and moreover, which is locally constant around the cusps. For \(\Gamma= \Gamma_0(N)\), where \(N\in \mathbb{F}_q [T]\) is a nonzero polynomial, the author proves that this Green's function can be described by Eisenstein series.
scientific article; zbMATH DE number 1375105