scholarly journals CONGRUENCES FOR THE COEFFICIENTS OF WEAKLY HOLOMORPHIC MODULAR FORMS

2006 ◽  
Vol 93 (2) ◽  
pp. 304-324 ◽  
Author(s):  
STEPHANIE TRENEER
Keyword(s):  
Partition Function ◽  
Wide Class ◽  
Modular Forms ◽  
Congruence Subgroup ◽  
Partition Functions ◽  
Modular Functions ◽  
Congruence Subgroups ◽  
Weakly Holomorphic Modular Forms ◽  
Linear Congruences ◽  
Holomorphic Modular Forms

Recent works have used the theory of modular forms to establish linear congruences for the partition function and for traces of singular moduli. We show that this type of phenomenon is completely general, by finding similar congruences for the coefficients of any weakly holomorphic modular form on any congruence subgroup $\Gamma_0 (N)$. In particular, we give congruences for a wide class of partition functions and for traces of CM values of arbitrary modular functions on certain congruence subgroups of prime level.

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

2014 ◽  
Vol 158 (1) ◽  
pp. 111-129 ◽  
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.

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

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.

A REMARK ON PRIME DIVISORS OF PARTITION FUNCTIONS

2014 ◽  
Vol 10 (01) ◽  
pp. 125-131
Author(s):  
PAUL POLLACK
Keyword(s):  
Partition Function ◽  
Wide Class ◽  
Analogous Result ◽  
Partition Functions ◽  
Prime Divisors ◽  
Classical Partition Function

Schinzel showed that the set of primes that divide some value of the classical partition function is infinite. For a wide class of sets 𝒜, we prove an analogous result for the function p𝒜(n) that counts partitions of n into terms belonging to 𝒜.

scholarly journals Lessons from the Ramond sector

2020 ◽  
Vol 9 (5) ◽  
Author(s):  
Nathan Benjamin ◽  
Ying-Hsuan Lin
Keyword(s):  
Partition Function ◽  
Field Theories ◽  
Partition Functions ◽  
Modular Invariance ◽  
Conformal Field Theories ◽  
Modular Functions ◽  
Conformal Field ◽  
Underlying Theory ◽  
Rule Out

We revisit the consistency of torus partition functions in (1+1)d fermionic conformal field theories, combining old ingredients of modular invariance/covariance with a modernized understanding of bosonization/fermionization dualities. Various lessons can be learned by simply examining the oft-ignored Ramond sector. For several extremal/kinky modular functions in the bootstrap literature, we can either rule out or identify the underlying theory. We also revisit the N = 1 Maloney-Witten partition function by calculating the spectrum in the Ramond sector, and further extending it to include the modular sum of seed Ramond characters. Finally, we perform the full N = 1 RNS modular bootstrap and obtain new universal results on the existence of relevant deformations preserving different amounts of supersymmetry.

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