On $p$-adic families of special elements for rank-one motives

TL;DR

This paper proposes a conjectural framework linking special elements of rank-one motives to Rubin-Stark elements via a higher-rank twist, offering a strategy to approach the Tamagawa Number Conjecture and providing new evidence in the CM abelian variety setting.

Contribution

It introduces a novel conjecture connecting special elements and Rubin-Stark elements through a higher-rank twist, advancing the understanding of the Tamagawa Number Conjecture.

Findings

The conjecture unifies known results and predictions.

A new strategy for proving the Tamagawa Number Conjecture is proposed.

Supporting evidence is provided in the context of CM abelian varieties.

Abstract

We conjecture that special elements associated with rank-one motives are obtained $p$-adically from Rubin-Stark elements by means of a precise `higher-rank Soul\'e twist' construction. We show this conjecture incorporates a variety of known results and existing predictions and also gives rise to a concrete strategy for proving the equivariant Tamagawa Number Conjecture for rank-one motives. We then use this approach to obtain new evidence in support of the equivariant Tamagawa Number Conjecture in the setting of CM abelian varieties.