Vector-valued modular forms and the Mock Theta Conjectures

2016/04/18 by Andersen, Nickolas · 1 citation
#11F37 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1604.05294

Abstract

The mock theta conjectures are ten identities involving Ramanujan's fifth-order mock theta functions. The conjectures were proven by Hickerson in 1988 using q-series methods. Using methods from the theory of harmonic Maass forms, specifically work of Zwegers and Bringmann-Ono, Folsom reduced the proof of the mock theta conjectures to a finite computation. Both of these approaches involve proving the identities individually, relying on work of Andrews-Garvan. Here we give a unified proof of the mock theta conjectures by realizing them as an equality between two nonholomorphic vector-valued modular forms which transform according to the Weil representation. We then show that the difference of these vectors lies in a zero-dimensional vector space.

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• Transcendental formulas for the coefficients of Ramanujan's mock theta functions

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