# The Shintani–Faddeev modular cocycle

# The Shintani–Faddeev modular cocycle

**Meeting ID: 920 2195 5230**

**Passcode: The three-digit integer that is the cube of the sum of its digits.**

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 SL_2(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