Abstract :
We present a new version “at s=1” of Rubinʹs refined, higher order Stark conjecture at s=0 for an abelian extension of number fields (K. Rubin, 1996, Ann. Inst. Fourier46, No. 1, 33–62). The key idea is to introduce a formalism of “twisted zeta-functions” to replace the L-functions underlying Rubinʹs 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.
