Wannabe modularity, the Shintani-Faddeev cocycle, and Stark units
Heilbronn Number Theory Seminar
23rd February 2022, 4:00 pm – 5:00 pm
We define wannabe modular forms, functions on the upper half-plane transforming under the action of the modular group by a generalized factor of automorphy. We define a related notion of wannabe Jacobi forms, and we show that the q-Pochhammer symbol is a meromorphic wannabe Jacobi form. Its factor of automorphy is the Shintani-Faddeev cocycle, 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 cocycle to exponentials of certain derivative L-values, conjectured by Stark to be algebraic units generating abelian extensions of real quadratic fields.