On Stark elements of arbitrary weight and their $p$-adic families

Abstract

We develop a detailed arithmetic theory related to special values at arbitrary integers of the Artin $L$-series of linear characters. To do so we define canonical generalized Stark elements of arbitrary `rank' and `weight', thereby extending the classical theory of Rubin-Stark elements. We then formulate an extension to arbitrary weight of the refined version of the Rubin-Stark Conjecture that we studied in an earlier article and also show that generalized Stark elements constitute a $p$-adic family by formulating precise conjectural congruence relations between elements of differing weights. We prove both of these conjectures in several important cases.