Joint PU/IAS Number Theory

We ask the question, “how does the infinite $q$-Pochhammer symbol transform under modular transformations?” and connect the answer to that question to the Stark conjectures. The infinite $q$-Pochhammer symbol transforms by a generalized factor of automorphy, or modular $1$-cocycle, that is analytic on a cut complex plane. This “Shintani–Faddeev modular cocycle” is an ${\rm SL}_2(\mathbb{Z})$-parametrized family of functions generalizing Shintani’s double sine function and Faddeev’s noncompact quantum dilogarithm. We relate real multiplication values of the Shintani–Faddeev modular cocycle to exponentials of certain derivative $L$-values, conjectured by Stark to be algebraic units generating abelian extensions of real quadratic fields.