Verifying the Congruence Conjecture for Rubin-Stark Elements

TL;DR

This paper investigates the Congruence Conjecture related to Rubin-Stark elements, providing numerical evidence and verifying its validity in 48 cases involving real quadratic fields.

Contribution

It develops techniques to numerically verify the Congruence Conjecture and confirms its validity in multiple cases with real quadratic base fields.

Findings

Complete verification of the conjecture in 48 cases

Development of numerical techniques for conjecture testing

Support for the conjecture's validity in specific number field cases

Abstract

The `Congruence Conjecture' was developed by the second author in a previous paper. It provides a conjectural explicit reciprocity law for a certain element associated to an abelian extension of a totally real number field whose existence is predicted by earlier conjectures of Rubin and Stark. The first aim of the present paper is to design and apply techniques to investigate the Congruence Conjecture numerically. We then present complete verifications of the conjecture in 48 varied cases with real quadratic base fields.