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

TL;DR

This paper introduces a new interpretation of Stark units in real quadratic fields as values of a modular cocycle derived from the $q$-Pochhammer symbol, linking number theory, modular forms, and quantum functions.

Contribution

It provides a novel cohomological perspective on Stark units via a modular cocycle related to the $q$-Pochhammer symbol and refines Shintani's Kronecker limit formula.

Findings

Relates Stark units to real multiplication values of a modular cocycle.

Connects the cocycle to the Shintani--Barnes double sine function and Faddeev quantum dilogarithm.

Refines Shintani's Kronecker limit formula with cohomological invariants.

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.