Global distributions and special zeta values

Abstract

In this paper, we develop the theory of global distributions (i.e. distributions of global fields) and apply it to the study of special values of abelian L-functions of a number field and division points of rank one Drinfeld modules. We introduce the concept of \epsilon-distributions and give examples by \epsilon-partial zeta functions. We determine the ranks of level groups of various kinds of universal distributions of a global field k, such as universal \epsilon-, punctured, punctured even and odd distributions of k. We show the universality of several distributions derived from special values of the \epsilon-partial zeta functions by studying \mathbb{Q}-linear independence of some special values. We also propose a conjecture and a question about the universality of \epsilon-distributions of special values of \epsilon-partial zeta functions.