Eisenstein series, cotangent-zeta sums, knots, and quantum modular forms

Abstract: 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.

number theory

Audience: researchers in the topic

Heilbronn number theory seminar

Series comments: This is part of the University of Bristol's Heilbronn number theory seminar. If you wish to attend the talk (and are not a Bristolian), please register using this form or email us at bristolhnts@gmail.com with your name and affiliation (if any).

We will email out the link to all registered participants the day before.