UC San Diego

The Stark conjectures attempt to capture the leading terms at s=0 of the S-incomplete Artin L-functions attached to an abelian extension of number fields as the image under a regulator map of an evaluator built out of S-units. We introduce a new conjecture of Popescu, which extends Rubin's higher order of vanishing Stark-type conjecture by removing the hypothesis that S contains splitting primes. We prove that the evaluator attached to an extension K/k< /em> can be written as a linear combination of evaluators arising in subextensions which do have splitting primes, linking the original conjecture of Rubin with its extension. This allows a cohomological proof of the extended conjecture when the original is known for the subextensions and S has ̀ènough'' finite unramifying primes. We study extensions of exponent 2 where we prove Rubin's conjecture under the hypothesis that an auxiliary smoothing set T is sufficiently large, and achieve new partial results towards the conjecture in general for these extensions. The consequences of a Stark-type conjecture of Burns are studied, leading to weaker sufficient inequalities for extensions of prime exponent. In the appendix, we prove a series of equivalences for when a cyclic Kummer extension of K is central extension over k, which is an analogue of Coates' condition for achieving an abelian extension