On the coefficients of the Drinfeld modular equation
The Drinfeld modular equation is the subject of this paper: an algorithm for computing its coefficients and estimates for its maximum coefficient are presented. Let \({\mathbb{F}}_q\) be the finite field of \(q\) elements, \(K={\mathbb{F}}_q(T)\) the function field over \({\mathbb{F}}_q\) in the variable \(T\), and \(A={\mathbb{F}}_q[T]\). Denote by \(K_{\infty}\) the completion of \(K\) at \(\infty=(1/T)\), and by \(C\) the completion of the algebraic closure of \(K_{\infty}\). Let \(\text{{deg}}:K\to {\mathbb{Z}}\cup\{-\infty\}, x\mapsto\text{deg } x\) be the valuation of \(K\) associated with the infinite place \(\infty\), and let \(|\;|\) be the corresponding absolute value (normalized by \(|T|=q\)). Fix an extension of \(|\;|\) to \(C\), and let \(\nu\) be a valuation of \(C\) defined by \(\nu(x)=\text{{log}}|x|\). For \(z\in\Omega:=C\setminus K_{\infty}\), let \([z,1]=Az+A\) be the rank \(2\) \(A\)--lattice in \(C\). A Drinfeld module \(\phi^z\) of rank \(2\) is defined by \(\phi^z_T:=TX+g(z)X^q+\Delta(z)X^{q^2}\). The \(j\)--invariant of \(\phi^z\) is \(j(z):=g(z)^{q+1}/\Delta(z)\). For a Drinfeld module \(\phi^z\) with \(j\)--invariant \(j(z)\), there is a Drinfeld modular equation \(\Phi_n(X,Y) \in{\mathbb{Z}}[X,Y]\) such that \(\Phi_n(X,Y)=\Phi_n(Y,X)\) and \[ \Phi_n(X,j(z))=\prod_{\gamma\in S_n} (X-j(\gamma z)), \] where \[ S_n=\left\{\left( \begin{matrix} a & b\\ 0 & d \end{matrix} \right)\in\text{{GL}}_2(A)\mid a,d\text{ are monic},\;(a,b,d)=1,\;ad=n, \text{deg }b< \text{deg }d \right\}, \] and \(\#S_n=\psi(n)=|n|\prod_{p|n,\;p\text{ prime}}(1+{1\over |p|})\). Let \(A_n\) denote the maximum of the absolute value of the coefficients of \(\Phi_n(X,Y)\), and put \(B_n=\text{{log}}_q A_n\). Theorem 1. \(A_n\) (and hence \(B_n\)) is not bounded as \(|n|\to\infty\). Theorem 2. For a fixed \(n\), an upper bound for \(B_n\) is given by \[ B_n\leq q\psi(n)+ \Biggl( {|n|\over q}-\text{deg} n \Biggr){(q-1)\psi(n)((q-1)\psi(n)+1) \over 2}+(q+1)|n|\psi(n). \] Further, an algorithm for computing the coefficients of \(\Phi_p(X,Y)\) (\(p\) a prime element in \(A\)) is presented. This extends the results for the classical modular equations to the Drinfeld modular equations.
scientific article; zbMATH DE number 1061242