A stark conjecture ''over Z'' for abelian L-functions with multiple zeros

Suppose K/k is an abelian extension of number fields. Stark's conjecture predicts, under suitable hypotheses, the existence of a global unit epsilon of K such that the special values L'(chi, 0) for all characters chi of Gal/(K/k) can be expressed as simple linear combinations of the logarithms of the different absolute values of epsilon. In this paper we formulate an extension of this conjecture, to attempt to understand the values L((r))(chi, 0) when the order of vanishing r may be greater than one. This conjecture no longer predicts the existence of individual special global units, but rather of special elements in an exterior power of the Galois module of global units (or S-units). We also discuss connections between this conjecture, class number formulas, and Euler systems.