Vector-valued modular forms and the Mock Theta Conjectures

TL;DR

This paper provides a unified proof of Ramanujan's mock theta conjectures by representing them as equalities between vector-valued modular forms and demonstrating their difference is zero.

Contribution

It introduces a novel approach using vector-valued modular forms and Weil representation to prove the mock theta conjectures collectively.

Findings

Unified proof of all ten mock theta conjectures.

Representation of identities as vector-valued modular form equalities.

Difference of the modular form vectors is zero, confirming the identities.

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.