Ken Ono

Affiliation: Emory University

Email: ono@mathcs.emory.edu

Title Of Talk: Ramanujan's last prophecy: quantum modular forms

URL:

Abstract: ``I discovered very interesting functions recently which I call ``Mock" theta-functions. Unlike the ``False" $\vartheta$-functions (studied partially by Prof. Rogers in his interesting paper) they enter into mathematics as beautifully as the ordinary theta functions." These are Ramanujan's words from his famous deathbed letter. Apart from a list of examples, including $f(q)$, the letter is a discussion of the claim: As $q$ approaches an even order $2k$ root of unity, we have $$ f(q)-(-1)^k (1-q)(1-q^3)(1-q^5)\cdots\left(1-2q+2q^4-\cdots\right)=O(1). $$ We prove this as a special case of a general result. The $O(1)$ constants are not mysterious; they are values of a ``quantum'' $q$-hypergeometric function which underlies a deep relationship between Dyson's rank mock theta function and the Andrews-Garvan crank modular form. Along these lines, we also prove that the Rogers-Fine false theta-functions specialize to quantum modular forms. Therefore, contrary to Ramanujan's claim, these functions do enter into mathematics as beautifully as the ordinary theta functions, and his own mock functions play a key role. This is joint work with Amanda Folsom and Rob Rhoades.

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