Drinfeld modular polynomials of level $T$

TL;DR

This paper studies Drinfeld modular polynomials related to $T$-isogenies between Drinfeld modules, providing explicit bounds and formulas for their coefficients' degrees, with numerical evidence supporting the sharpness of these bounds.

Contribution

It offers an explicit classification of $T$-isogenies and derives precise bounds and formulas for the degrees of the modular polynomial coefficients.

Findings

Derived explicit bounds on the degrees of modular polynomial coefficients.

Obtained exact formulas for the height of modular polynomials.

Numerical computations confirm the bounds are often sharp.

Abstract

We investigate Drinfeld modular polynomials parametrizing $T$-isogenies between Drinfeld $\mathbb{F}_q[T]$-modules of rank $r\geq 2$. 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.