Distributive properties of division points and discriminants of Drinfeld modules

Abstract

We present a new notion of distribution and derived distribution of rank r ∈ 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 μ(K) of roots of unity in the algebraic closure K of K, and in the group U(1)(C∞) of 1-units of the completed algebraic closure C∞ of K∞, respectively. There result product formulas for division points and discriminants that encompass known results (e.g. analogues of Wallis’ formula for (2πı)2 in the rank-1 case, of Jacobi’s formula Δ = (2πı)12q (1−qn)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, ∞)=(Fq(T),∞) and r = 1, 2 or 3, we give explicit values for the logarithms of such forms.