CLASSIFICATION OF CONGRUENCES FOR MOCK THETA FUNCTIONS AND WEAKLY HOLOMORPHIC MODULAR FORMS

AbstractWe show that someq-series such as universal mock theta functions are linear sums of theta quotients and mock Jacobi forms of weight 1/2, which become holomorphic parts of real analytic modular forms when they are restricted to torsion points and multiplied by suitable powers ofq. We also prove that certain linear sums ofq-series are weakly holomorphic modular forms of weight 1/2 due to annihilation of mock Jacobi forms or completion by mock Jacobi forms. As an application, we obtain a relation between the rank and crank of a partition.

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Mock theta functions and weakly holomorphic modular forms modulo 2 and 3

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004114000565 ◽

2014 ◽

Vol 158 (1) ◽

pp. 111-129 ◽

Cited By ~ 4

Author(s):

SCOTT AHLGREN ◽

BYUNGCHAN KIM

Keyword(s):

Partition Function ◽

Wide Class ◽

Modular Forms ◽

Theta Functions ◽

Mock Theta Functions ◽

Weakly Holomorphic Modular Forms ◽

Linear Congruences ◽

Image Position ◽

Holomorphic Modular Forms

AbstractWe prove that the coefficients of the mock theta functions \begin{eqnarray*} f(q) = \sum_{n=1}^{\infty} \frac{ q^{n^2}}{(1+q)^2 (1+q^2)^2 \cdots (1+q^n)^2 } \end{eqnarray*} and \begin{eqnarray*} \omega(q)=1+\sum_{n=1}^\infty \frac{q^{2n^2+2n}}{(1+q)^2(1+q^3)^2\cdots (1+q^{2n+1})^2} \end{eqnarray*} possess no linear congruences modulo 3. We prove similar results for the moduli 2 and 3 for a wide class of weakly holomorphic modular forms and discuss applications. This extends work of Radu on the behavior of the ordinary partition function.

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Higher depth mock theta functions and q-hypergeometric series

Forum Mathematicum ◽

10.1515/forum-2021-0013 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Joshua Males ◽

Andreas Mono ◽

Larry Rolen

Keyword(s):

Theta Function ◽

Modular Forms ◽

Hypergeometric Series ◽

Theta Functions ◽

Mock Theta Function ◽

Mock Theta Functions ◽

Maass Forms ◽

Mock Modular Forms ◽

Harmonic Maass Forms

Abstract In the theory of harmonic Maaß forms and mock modular forms, mock theta functions are distinguished examples which arose from q-hypergeometric examples of Ramanujan. Recently, there has been a body of work on higher depth mock modular forms. Here, we introduce distinguished examples of these forms, which we call higher depth mock theta functions, and develop q-hypergeometric expressions for them. We provide three examples of mock theta functions of depth two, each arising by multiplying a classical mock theta function with a certain specialization of a universal mock theta function. In addition, we give their modular completions, and relate each to a q-hypergeometric series.

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On some automorphic properties of Galois traces of class invariants from generalized Weber functions of level 5

Open Mathematics ◽

10.1515/math-2019-0115 ◽

2019 ◽

Vol 17 (1) ◽

pp. 1631-1651

Author(s):

Ick Sun Eum ◽

Ho Yun Jung

Keyword(s):

Modular Forms ◽

Fourier Coefficients ◽

Singular Values ◽

Congruence Subgroups ◽

Maass Form ◽

Weakly Holomorphic Modular Forms ◽

Reciprocity Law ◽

Class Invariants ◽

Higher Genus ◽

Holomorphic Modular Forms

Abstract After the significant work of Zagier on the traces of singular moduli, Jeon, Kang and Kim showed that the Galois traces of real-valued class invariants given in terms of the singular values of the classical Weber functions can be identified with the Fourier coefficients of weakly holomorphic modular forms of weight 3/2 on the congruence subgroups of higher genus by using the Bruinier-Funke modular traces. Extending their work, we construct real-valued class invariants by using the singular values of the generalized Weber functions of level 5 and prove that their Galois traces are Fourier coefficients of a harmonic weak Maass form of weight 3/2 by using Shimura’s reciprocity law.

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