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

Abstract

In this article we shall generalize the theory of Stark elements to arbitrary ‘weight’. This means that we define canonical generalized Stark elements that correspond to the leading terms at arbitrary integers of $L$-series attached to the multiplicative group. We investigate their basic properties, and formulate both a conjecture on the precise integrality condition that these Stark elements satisfy and a conjecture that describes the Galois module structures of certain cohomology groups as a generalization of the refined Rubin-Stark conjecture we formulated in an earlier article ‘On zeta elements for $\mathbb{G}_m$’ in Doc. Math. 21 (2016). We then prove these conjectures in several interesting cases. The integrality condition studied here will then play an important role in a subsequent article of the first and third authors which formulates a precise conjectural congruence relation between the generalized Stark elements of differing weights that are defined here.