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

TL;DR

This paper extends the classical Stark conjectures by defining generalized Stark elements of arbitrary weight and rank, exploring their $p$-adic families, and proving key conjectures in significant cases.

Contribution

It introduces a new framework for Stark elements of arbitrary weight and rank, extending Rubin-Stark theory and formulating conjectural congruences in $p$-adic families.

Findings

Formulated an extension of the Rubin-Stark Conjecture to arbitrary weight.

Proved the conjectures in several important cases.

Established that generalized Stark elements form a $p$-adic family.

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.