Global distributions and special zeta values

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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.