The fundamental domain of the tree of \(GL(2)\) over the function field of an elliptic curve (Q1320627)

Let \(E\) be an elliptic curve over a field \(k\), and \(k[E]\) be its affine coordinate ring. If \(t\) is a local uniformizer at \(\infty\), then one can embed \(k[E]\) in \(k((t))\). Let \(\Gamma = GL(2,k[E])\), \(K = GL(2,k [[T]])\), \(G = GL(2,k ((t)))\) and \(Z\) be the center of \(G\). Then, following \textit{J.-P. Serre} [``Trees'' (Berlin 1980; Zbl 0548.20018)], one can define a tree structure \(T\) on \(G/KZ\), and the quotient graph \(\Gamma \backslash T\) is well defined. The aim of this paper is to determine the shape of \(\Gamma \backslash T\). More specifically, the author defines a subtree \(S\) of \(T\), such that \(S \approx \Gamma \backslash T\).