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The Shintani--Faddeev modular cocycle: Stark units from $q$-Pochhammer ratios

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arxiv 2411.06763 v3 pith:2SC263XR submitted 2024-11-11 math.NT

classification math.NT
keywords cocyclemodularrealstarkinvariantsmultiplicationpochhammerunits
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abstract

We give a new interpretation of Stark units associated to real quadratic fields as real multiplication values of a modular cocycle. The cocycle of interest is a meromorphic factor describing the modular transformations of the $q$-Pochhammer symbol and is related to the Shintani--Barnes double sine function and the Faddeev quantum dilogarithm. We prove a refinement of Shintani's Kronecker limit formula that relates square roots of Stark class invariants to real multiplication values of the cocycle, which are cohomological invariants.

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