A family of mock theta functions of weights 1/2 and 3/2 and their congruence properties

2014/02/26 by René Olivetto, Olivetto, René
Mathematics · #11F33 #11F37 #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT) #math.NT #msc:11F33 #msc:11F37

paper · pdf · doi:10.48550/arxiv.1402.6509

arxiv created 2014/02/26 · openalex publication_date 2014/02/26 · arxiv updated 2014/02/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In a private communication, K. Ono conjectured that any mock theta function of weight 1/2 or 3/2 can be congruent modulo a prime p to a weakly holomorphic modular form for just a few values of p. In this paper we describe when such a congruence occurs. More precisely we show that it depends on the p-adic valuation of the mock theta function itself. In order to do so, we construct a family of mock theta functions in terms of derivatives of the Appell sum, which have a special Fourier expansion at infinity.

Citations

• Mock Theta Functions
• Dyson’s ranks and Maass forms
• On two geometric theta lifts
• Congruence properties for the partition function

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• A Unified Partial and Mock Theta Function
• Radial limits of mock theta functions
• On certain explicit congruences for mock theta functions
• Mock Modular Forms with Integral Fourier Coefficients