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.
Highlights
1 Background In his last letter to Hardy, dated three months before his death in early 1920, Ramanujan briefly described a new class of functions which he called mock theta functions, and he listed 17 examples [2, p. 220]
The fifth order mock theta functions he further divided into two groups1; for example, four of these fifth order functions are f0(q) qn2 (−q; q)n q2n2 (q; q2)n
According to Gordon and McIntosh [14, p. 106], the mock theta conjectures together form “one of the fundamental results of the theory of [mock theta functions]” and Hickerson’s proof is a “tour de force.”. In his Ph.D. thesis [23], Zwegers showed that the mock theta functions can be completed to real analytic modular forms of weight 1/2 by multiplying by a suitable rational power of q and adding nonholomorphic integrals of certain unary theta series of weight 3/2
Summary
In his last letter to Hardy, dated three months before his death in early 1920, Ramanujan briefly described a new class of functions which he called mock theta functions, and he listed 17 examples [2, p. 220]. 106], the mock theta conjectures together form “one of the fundamental results of the theory of [mock theta functions]” and Hickerson’s proof is a “tour de force.” In his Ph.D. thesis [23], Zwegers showed that the mock theta functions can be completed to real analytic modular forms of weight 1/2 by multiplying by a suitable rational power of q and adding nonholomorphic integrals of certain unary theta series of weight 3/2. Zagier makes a similar comment in [22, §6] Following their approach, Folsom [11] reduced the proof of the χ0(q) and χ1(q) mock theta conjectures to the verification of two identities in the space of modular forms of weight 1/2 for the subgroup G = 1(144 · 102 · 54).
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