On plane models for Drinfeld modular curves
Notation: \(A = {\mathbb F}_q[T]\) denotes the polynomial ring in one variable \(T\) over a field with \(q\) elements; \(C\) denotes the completion of the algebraic closure of the local field \({\mathbb F}_q((1/T))\). Let \(f\) be a non-zero element of \(A\). The (coarse) moduli space \(Y_0(f)\) of rank \(2\) Drinfeld \(A\)-modules over \(C\) together with a sub-module isomorphic with \(A/f\) can be described analytically by \(Y(f)(C) = \Omega / \Gamma_0(f)\) where \(\Omega := \mathbb{P}^1 (C) - \mathbb{P}^1 ({\mathbb F}_q((1/T)))\) is the Drinfeld upper half plane and \(\Gamma(f)\) is the subgroup of \(\text{GL}(2, A)\) consisting of those matrices whose left bottom entry is divisible by \(f\). \(Y_0(f)\) is a smooth affine algebraic curve over \(C\). One method to obtain explicit equations for its function field is to use the theory of Drinfeld modular forms to construct two meromorphic functions on the completion \(X_0(f)\) of \(Y_0(f)\) that generate the function field, and to find the algebraic relation that they satisfy. In this paper the authors consider the case where \(f = gh\) with \(g\) irreducible of degree \(1\) or \(2\) and \(h\) any polynomial that is coprime with \(g\). Using the fact that \(X_0(g)\) is rational they exhibit such a pair of generators and the relation between them. Then two explicit examples (with \(q = 2\) and \(f\) of degree \(2\) and \(3\)) are presented. It turns out that the equations obtained for \(X_0(f)\) have considerably smaller coefficients than those computed previously by \textit{A. Schweizer} [J. Number Theory 52, No. 1, 53--68 (1995; Zbl 0826.11026)]. Finally the authors use singular values of these meromorphic functions to generate certain ring class fields.
scientific article; zbMATH DE number 5072780