On the coefficients of modular polynomials for Drinfeld modules
Let \(h(\phi)\) be the logarithmic height of the polynomial \(\phi\) (i.e., the maximum of logarithms of absolute values of the coefficients of \(\phi)\). A result of \textit{P. Cohen} [Math. Proc. Camb. Philos. Soc. 95, 389-402 (1984; Zbl 0541.10026)] states that the modular polynomial \(\phi_m\) \((m\in\mathbb{N})\) of elliptic curves satisfies \[ h(\phi_m)= 6\psi(m)(\log m- 2\kappa(m)+ O(1)) \] with well-known arithmetic functions \(\psi(m)\) and \(\kappa(m)\). Following the analogy between elliptic curves and Drinfeld \(A\)-modules of rank two (here \(A\) is the polynomial ring \(\mathbb{F}_q[T]\) over a finite field \(\mathbb{F}_q\)), the author proves a similar statement for the height of the Drinfeld modular polynomial \(\phi_m(x,y)\) (where \(m\in A\) is a non-constant monic polynomial over \(\mathbb{F}_q\)). Main ingredients are Farey sequences for \(A\) (the theory of which is partially developed), estimates on the Drinfeld invariant \(j(t)\) in terms of the imaginary distance \(| z|_i\), and some non-archimedean function theory.
scientific article; zbMATH DE number 1236189