Classification of congruences for mock theta functions and weakly holomorphic modular forms
Let \(p(n)\) denote the number of partitions of \(n\). \textit{C.-S. Radu} [Trans. Am. Math. Soc. 365, No. 9, 4881--4894 (2013; Zbl 1336.11071)] has shown that if the congruence NEWLINE\[NEWLINE p(mn + t) \equiv 0 \pmod{\ell} NEWLINE\]NEWLINE holds for some prime \(\ell \geq 5\), then \(\ell \mid m\) and \(\left(\frac{24t-1}{\ell}\right) \neq \left(\frac{-1}{\ell}\right)\). The author shows that the same is true with \(p(n)\) replaced by \(a(n)\), where \(a(n)\) is the coefficient of \(q^n\) in Ramanujan's third order mock theta function \(f(q)\), NEWLINE\[NEWLINE f(q) = \sum_{n \geq 0} \frac{q^{n^2}}{(1+q)^2(1+q^2)^2\cdots(1+q^n)^2}. NEWLINE\]NEWLINE He also shows that a similar result holds for the third order mock theta function NEWLINE\[NEWLINE \omega(q) = 1+ \sum_{n \geq 1} \frac{q^{2n^2+2n}}{(1+q)^2(1+q^3)^2\cdots(1+q^{2n-1})^2} NEWLINE\]NEWLINE as well as for functions of the form NEWLINE\[NEWLINE f(z) = \prod_{\delta \mid N}\eta(\delta z)^{r_{\delta}}, NEWLINE\]NEWLINE where \(\eta(z)\) is the usual Dedekind \(\eta\)-function.NEWLINENEWLINEThe proofs follow the methods of Radu. They depend on results of Deligne and Rapoport relating the expansion of a modular form at \(\infty\) to its expansion at other cusps as well as the transformation properties of the relevant (mock) modular forms restricted to arithmetic progressions. In the cases considered here, these transformation properties were determined by \textit{S. Ahlgren} and \textit{B. Kim} [``Mock theta functions and weakly holomorphic modular forms modulo 2 and 3'', Preprint, \url{arXiv:1306.4386}].
scientific article; zbMATH DE number 6353661
scientific article