Distributive properties of division points and discriminants of Drinfeld modules

Abstract

We present a new notion of distribution and derived distribution of rank $r\in N$ for a global function field K with a distinguished place ∞. It allows to describe the relations between division points, isogenies, and discriminants both for a fixed Drinfeld module of rank r for the above data, or for the corresponding modular forms.

We introduce and study three basic distributions with values in $Q$, in the group $\mu \left(\bar{K}\right)$ of roots of unity in the algebraic closure $\bar{K}$ of K, and in the group $U^{\left(1\right)}\left(C_{\infty }\right)$ of 1-units of the completed algebraic closure $C_{\infty }$ of $K_{\infty }$, respectively.

There result product formulas for division points and discriminants that encompass known results (e.g. analogues of Wallis' formula for ${\left(2\pi ı\right)}^2$ in the rank-1 case, of Jacobi's formula $\Delta ={\left(2\pi ı\right)}^{12}q\prod {(1-q^n)}^{24}$ in the rank-2 case, and similar boundary expansions for $r>2$) and several new ones: the definition of a canonical discriminant for the most general case of Drinfeld modules and the description of the sizes of division and discriminant forms.

In the now classical case where $(K,\infty )=\left(F_q\right(T),\infty )$ and $r=1$, 2 or 3, we give explicit values for the logarithms of such forms.