Heights of Drinfeld modular polynomials and Hecke images

Abstract

We obtain explicit upper and lower bounds on the size of the coefficients of the Drinfeld modular polynomials ${\Phi }_N$ for any monic $N\in {\mathbb{F}}_q[t]$. These polynomials vanish at pairs of $j$-invariants of Drinfeld ${\mathbb{F}}_q[t]$-modules of rank 2 linked by cyclic isogenies of degree $N$. The main term in both bounds is asymptotically optimal as $\mathrm{deg}(N)$ tends to infinity. We also obtain precise estimates on the Weil height and Taguchi height of Hecke images of Drinfeld modules of rank 2.