A closed form expression for the Drinfeld modular polynomial ${\Phi }_T(X,Y)$

Abstract

In this paper we give a closed form expression for the Drinfeld modular polynomial ${\Phi }_T(X,Y)\in {\mathbb{F}}_q(T)[X,Y]$ for arbitrary $q$ and prove a conjecture of Schweizer. A new identity involving the Catalan numbers plays a central role.