A closed form expression for the Drinfeld modular polynomial $\Phi_T(X,Y)$

TL;DR

This paper derives a closed form expression for the Drinfeld modular polynomial _T(X,Y) for any q, confirming Schweizer's conjecture and introducing a new Catalan number identity.

Contribution

It provides the first explicit closed form for _T(X,Y) for all q and proves a related conjecture, advancing understanding of Drinfeld modular forms.

Findings

Closed form expression for _T(X,Y) for arbitrary q

Proof of Schweizer's conjecture

A new identity involving Catalan numbers

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.