Twisted zeta-functions and abelian Stark Conjectures

We present a new version "at s = 1" of Rubin's refined, higher order Stark conjecture at s = 0 for an abelian extension of number fields (K. Rubin, 1996, Ann. Inst. Fourier 46, No. 1, 33-62). The key idea is to introduce a formalism of "twisted zeta-functions" to replace the L-functions underlying Rubin's conjecture. This achieves certain simplifications, notably eliminating Gauss sums in a natural way from our version at s = 1. It also facilitates some further developments, including an important motivation of the present paper: the formulation of an analogous p-adic conjecture to be presented in a sequel. (C) 2002 Elsevier Science (USA).