Heights of Drinfeld modular polynomials and Hecke images

TL;DR

This paper provides explicit bounds on the coefficients of Drinfeld modular polynomials and estimates on heights of Hecke images, advancing understanding of their size and arithmetic properties in the function field setting.

Contribution

It introduces explicit bounds for Drinfeld modular polynomial coefficients and precise height estimates for Hecke images, improving previous asymptotic results.

Findings

Bounds on coefficients are asymptotically optimal as degree increases.

Explicit height estimates for Hecke images of rank 2 Drinfeld modules.

Main term in bounds matches asymptotic behavior for large degrees.

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.