Eisenstein series, cotangent-zeta sums, knots, and quantum modular forms
Heilbronn Number Theory Seminar
2nd December 2020, 4:00 pm – 5:00 pm
Quantum modular forms, defined in the rational numbers, transform like modular forms do on the upper half-plane, up to suitably analytic error functions. In this talk we give frameworks for two different examples of quantum modular forms originally due to Zagier: the Dedekind sum, and a certain q-hypergeometric sum due to Kontsevich. For the first, we extend work of Bettin and Conrey and define twisted Eisenstein series, study their period functions, and establish quantum modularity of certain cotangent-zeta sums. For the second, we discuss results due to Hikami, Lovejoy, the author, and others, on quantum modular and quantum Jacobi forms related to colored Jones polynomials for certain families of knots.