Testing the Congruence Conjecture for Rubin–Stark elements

The ‘Congruence Conjecture’ was developed by the second author in a previous paper [So3]. 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.