Results 11 to 20 of about 205 (137)
Congruences involving the $U_{\\ell}$ operator for weakly holomorphic\n modular forms [PDF]
The Ramanujan Journal, 2019 Let $ $ be an integer, and $f(z)=\sum_{n\gg-\infty} a(n)q^n$ be a weakly holomorphic modular form of weight $ +\frac 12$ on $ _0(4)$ with integral coefficients. Let $\ell\geq 5$ be a prime. Assume that the constant term $a(0)$ is not zero modulo $\ell$. Further, assume that, for some positive integer $m$, the Fourier expansion of $(f|U_{\ell^m})(z) =
Dohoon Choi, Subong Lim
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CONGRUENCES FOR THE COEFFICIENTS OF WEAKLY HOLOMORPHIC MODULAR FORMS [PDF]
Proceedings of the London Mathematical Society, 2006 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 $\Gamma_0 (N)$.
Stephanie Treneer
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Weakly holomorphic modular forms for some moonshine groups [PDF]
Archiv der Mathematik, 2014 In an article in the Pure and Applied Mathematics Quarterly in 2008, Duke and Jenkins investigated a certain natural basis of the space of weakly holomorphic modular forms for the full modular group $SL_2({\bf Z})$. We show here that their results can be generalized to certain moonshine groups, also allowing characters that are real on the underlying ...
Martina Lahr, Rainer Schulze‐Pillot
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A Basis for the space of weakly holomorphic Drinfeld modular forms of level $T$ [PDF]
Journal of Number Theory, 2023 In this article, we explicitly construct a canonical basis for the space of certain weakly holomorphic Drinfeld modular forms for $Γ_0(T)$ (resp., for $Γ_0^+(T)$) and compute the generating function satisfied by the basis elements. We also give an explicit expression for the action of the $Θ$-operator, which depends on the divisor of meromorphic ...
Tarun Dalal
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Rank generating functions as weakly holomorphic modular forms [PDF]
Acta Arithmetica, 2008 We study infinite families of generating functions involving the rank of the ordinary partition function, which include as special cases many of the generating functions introduced by Atkin and Swinnerton-Dyer in the 1950s. We prove that each of these generating functions is a weakly holomorphic modular form of weight 1/2 on some congruence subgroup Γ1(
Scott Ahlgren, Stephanie Treneer
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On the zeros of weakly holomorphic modular forms [PDF]
Archiv der Mathematik, 2014 In this article, we study the nature of zeros of weakly holomorphic modular forms. In particular, we prove results about transcendental zeros of modular forms of higher levels and for certain Fricke groups which extend a work of Kohnen. Furthermore, we investigate the algebraic independence of values of weakly holomorphic modular forms.
Sanoli Gun, Biswajyoti Saha
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A note on congruences for weakly holomorphic modular forms [PDF]
Proceedings of the American Mathematical Society, 2021 Let O L O_L be the ring of integers of a number field L L . Write q = e 2 π i z q = e^{2 \pi i z} , and suppose that f ( z ) = ∑
Spencer Dembner, Vanshika Jain
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CLASSIFICATION OF CONGRUENCES FOR MOCK THETA FUNCTIONS AND WEAKLY HOLOMORPHIC MODULAR FORMS [PDF]
The Quarterly Journal of Mathematics, 2013 Let $f(q)$ denote Ramanujan's mock theta function \[f(q) = \sum_{n=0}^{\infty} a(n) q^{n} := 1+\sum_{n=1}^{\infty} \frac{q^{n^{2}}}{(1+q)^{2}(1+q^{2})^{2}\cdots(1+q^{n})^{2}}.\] It is known that there are many linear congruences for the coefficients of $f(q)$ and other mock theta functions. We prove that if the linear congruence $a(mn+t) \equiv 0 \pmod{
Nickolas Andersen
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On the Zeros and Coefficients of Certain Weakly Holomorphic Modular Forms [PDF]
Pure and Applied Mathematics Quarterly, 2008 A \textit{weakly holomorphic modular form} (say, \(f\)) of weight \(k\in 2\mathbb{Z}\) for the full modular group \(\mathrm{PSL}_{2}(\mathbb{Z})\) is defined the same way as holomorphic modular form, only \(f\) is allowed to have a finite number of negative powers in its \(q\)-expansion. Write \(k=12\ell+k'\) with uniquely determined \(\ell\in\mathbb{Z}
William Duke, Paul A. Jenkins
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Hecke grids and congruences for weakly holomorphic modular forms [PDF]
, 2013 Let $U(p)$ denote the Atkin operator of prime index $p$. Honda and Kaneko proved infinite families of congruences of the form $f|U(p) \equiv 0 \pmod{p}$ for weakly holomorphic modular forms of low weight and level and primes $p$ in certain residue classes, and conjectured the existence of similar congruences modulo higher powers of $p$. Partial results
Scott Ahlgren, Nickolas Andersen
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