Drinfeld modular polynomials of level $T$

We investigate Drinfeld modular polynomials parametrizing $T$-isogenies between Drinfeld ${\mathbb{F}}_q[T]$-modules of rank $r\ge 2$. By providing an explicit classification of such isogenies, we derive explicit bounds on the $T$-degrees of the coefficients of the associated modular polynomials. In particular, we obtain exact expressions for the height (i.e. degree of the largest coefficient) of these modular polynomials. Numerical computations show that the bounds on the smaller coefficients are often sharp, too.