Distributive properties of division points and discriminants of Drinfeld modules

Abstract

We present a new notion of distribution and derived distribution of rank $r \in \mathbb{N}$ for a global function field $K$ with a distinguished place $\infty$. 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 $\mathbb{Q}$, in the group $\mu(\overline{K})$ of roots of unity in the algebraic closure $\overline{K}$ of $K$, and in the group $U^{(1)}(C_{\infty})$ 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 $(2\pi i)^{2}$ in the rank-$1$ case, of Jacobi's formula $\Delta = (2\pi i)^{12} q \prod (1-q^{n})^{24}$ 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 $(K, \infty) = (\mathbb{F}_{q}(T), \infty)$ and $r = 1$, $2$ or $3$, we give explicit values for the logarithms of such forms.