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The values of the so-called {\em Dedekind--Rademacher cocycle} at certainreal quadratic arguments are shown to be global $p$-units in the narrow Hilbertclass field of the associated real quadratic field, as predicted by conjecturesof Darmon, Dasgupta, and Vonk. The strategy for proving this result combines anapproach of Darmon-Pozzi-Vonk with one crucial extra ingredient: the study ofinfinitesimal deformations of irregular Hilbert Eisenstein series of weight onein the anti-parallel direction, building on the techniques in earlier work ofBetina, Dimitrov, and Pozzi.

The values of the Dedekind-Rademacher cocycle at real multiplication points