Harmonic Weak Maass Forms Research Articles

Generalisation of the Asai-Kaneko-Ninomiya identity to higher level

Let j be the classical modular invariant, which is a modular function of weight zero for SL2(Z). Then for every z,τ in the upper half-plane, the Asai-Kaneko-Ninomiya (AKN) identity relates j(z)−j(τ) to the generating function of the Hecke system jm(τ) of modular functions for SL2(Z). In [7] the modular functions jm(z) were generalised to modular functions JN,m for higher level using harmonic weak Maass forms. In this paper, we show that under certain conditions on JN,m, the AKN identity can be generalised to higher level Γ0(N) using Poincaré series of weight 2 and level N. We also prove an infinite product formula for JN,1 which generalises the denominator formula for the Monster Lie algebra.

Harmonic weak Maass forms and periods II

Harmonic weak Maass forms and periods II

On Kleinian mock modular forms

We give an explicit and computationally efficient construction of harmonic weak Maass forms which map to weight 2 newforms under the ξ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi $$\\end{document}-operator. Our work uses a new non-analytic completion of the Kleinian ζ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\zeta $$\\end{document}-function from the theory of Abelian functions.

Hecke equivariance of generalized Borcherds products of type O(2,1)

Recently, a weak converse theorem for Borcherds' lifting operator of type O(2,1) for Γ0(N) is proved and the logarithmic derivative of a modular form for Γ0(N) is explicitly described in terms of the values of Niebur-Poincaré series at its divisors in the complex upper half-plane. In this paper, we prove that the generalized Borcherds' lifting operator of type O(2,1) is Hecke equivariant under the extension of Guerzhoy's multiplicative Hecke operator on the integral weight meromorphic modular forms and the Hecke operator on half-integral weight vector-valued harmonic weak Maass forms. Additionally, we show that the logarithmic differential operator is also Hecke equivariant under the multiplicative Hecke operator and the Hecke operator on integral weight meromorphic modular forms. As applications of Hecke equivariance of the two operators, we obtain relations for twisted traces of singular moduli modulo prime powers and congruences for twisted class numbers modulo primes, including those associated to genus 1 modular curves.

Completions and algebraic formulas for the coefficients of Ramanujan\u2019s mock theta functions

We present completions of mock theta functions to harmonic weak Maass forms of weight nicefrac {1}{2} and algebraic formulas for the coefficients of mock theta functions. We give several harmonic weak Maass forms of weight nicefrac {1}{2} that have mock theta functions as their holomorphic part. Using these harmonic weak Maass forms and the Millson theta lift, we compute finite algebraic formulas for the coefficients of the appearing mock theta functions in terms of traces of singular moduli.

Bases of spaces of harmonic weak Maass forms and Shintani lifts of harmonic weak Maass forms

We construct bases of the space of harmonic weak Maass forms of weight $$\kappa \in \frac{1}{2}{\mathbb {Z}}$$ . Using these bases, we obtain a Shintani lift from a positive integral weight harmonic weak Maass form to a half-integral weight harmonic weak Maass form, which reduces to the classical Shintani lift on the space of cusp forms.

Modularity of Galois traces of ray class invariants

After Zagier’s significant work (in: Bogomolov and Katzarkov (eds) Motives, polylogarithms and hodge theory, part I, International Press, Somerville, 2002) on traces of singular moduli, Bruinier and Funke (J Reine Angew Math 594:1–33, 2006) generalized his result to the traces of singular values of modular functions on modular curves of arbitrary genus. In class field theory, the extended ring class field is a generalization of the ray class field over an imaginary quadratic field. By using Shimura’s reciprocity law, we construct primitive generators of the extended ring class fields by using Siegel functions of arbitrary level $$N\ge 2$$ and identify their Galois traces with Fourier coefficients of weight 3/2 harmonic weak Maass forms. This would extend the results of Jeon et al. (Math Ann 353:37–63, 2012) and Jung et al. (Modularity of Galois traces of Weber’s resolvents, under revision).

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

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.

Schneider–Siegel theorem for a family of values of a harmonic weak Maass form at Hecke orbits

Abstract Let j ⁢ ( z ) {j(z)} be the modular j-invariant function. Let τ be an algebraic number in the complex upper half plane ℍ {\mathbb{H}} . It was proved by Schneider and Siegel that if τ is not a CM point, i.e., [ ℚ ( τ ) : ℚ ] ≠ 2 {[\mathbb{Q}(\tau):\mathbb{Q}]\neq 2} , then j ⁢ ( τ ) {j(\tau)} is transcendental. Let f be a harmonic weak Maass form of weight 0 on Γ 0 ⁢ ( N ) {\Gamma_{0}(N)} . In this paper, we consider an extension of the results of Schneider and Siegel to a family of values of f on Hecke orbits of τ. For a positive integer m, let T m {T_{m}} denote the m-th Hecke operator. Suppose that the coefficients of the principal part of f at the cusp i ⁢ ∞ {i\infty} are algebraic, and that f has its poles only at cusps equivalent to i ⁢ ∞ {i\infty} . We prove, under a mild assumption on f, that, for any fixed τ, if N is a prime such that N ≥ 23 {N\geq 23} and N ∉ { 23 , 29 , 31 , 41 , 47 , 59 , 71 } {N\notin\{23,29,31,41,47,59,71\}} , then f ( T m . τ ) {f(T_{m}.\tau)} are transcendental for infinitely many positive integers m prime to N.

Pairs of eta-quotients with dual weights and their applications

Let D be the differential operator defined by D:=12πiddz. This induces a map Dk+1:M−k!(Γ0(N))→Mk+2!(Γ0(N)), where Mk!(Γ0(N)) is the space of weakly holomorphic modular forms of weight k on Γ0(N). The operator Dk+1 plays important roles in the theory of Eichler-Shimura cohomology and harmonic weak Maass forms. On the other hand, eta-quotients are fundamental objects in the theory of modular forms and partition functions. In this paper, we show that the structure of eta-quotients is very rarely preserved under the map Dk+1 between dual spaces M−k!(Γ0(N)) and Mk+2!(Γ0(N)). More precisely, we classify dual pairs (f,Dk+1f) under the map Dk+1 such that f is an eta-quotient and Dk+1f is a non-zero constant multiple of an eta-quotient. When the levels are square-free, we give the complete classification of such pairs. In general, we find a necessary condition for such pairs: the weight of the primitive eta-quotient f(z)=η(di1z)b1⋯η(ditz)bt is less than or equal to 4 and every prime divisor of each di is less than 11. We also give various applications of these classifications. In particular, we find all eta-quotients of weight 2 and square-free level N such that they are in the Eisenstein space for Γ0(N). To prove our main theorems, we use various combinatorial properties of a Latin square matrix whose rows and columns are exactly divisors of N.

Values of harmonic weak Maass forms on Hecke orbits

Let q:=e2πiz, where z∈H. For an even integer k, let f(z):=qh∏m=1∞(1−qm)c(m) be a meromorphic modular form of weight k on Γ0(N). For a positive integer m, let Tm be the mth Hecke operator and D be a divisor of a modular curve with level N. Both subjects, the exponents c(m) of a modular form and the distribution of the points in the support of Tm.D, have been widely investigated. When the level N is one, Bruinier, Kohnen, and Ono obtained, in terms of the values of j-invariant function, identities between the exponents c(m) of a modular form and the points in the support of Tm.D. In this paper, we extend this result to general Γ0(N) in terms of values of harmonic weak Maass forms of weight 0. By the distribution of Hecke points, this applies to obtain an asymptotic behavior of convolutions of sums of divisors of an integer and sums of exponents of a modular form.

Twisted Borcherds Products Associated to Genus Zero $$\Gamma _{0}(N)$$

In this short note, we realize a product associated to a uniformizer for genus zero $$\Gamma _{0}(N)$$ twisted with a genus character as a twisted Borcherds product lifted from a weight $$\frac{1}{2}$$ harmonic weak Maass form for the Weil representation.

On a theta lift related to the Shintani lift

We study the Millson theta lift which maps weight −2k to weight 1/2−k harmonic weak Maass forms for k∈Z,k≥0, and which is closely related to the classical Shintani lift from weight 2k+2 to weight k+3/2 cusp forms. We compute the Fourier expansion of the theta lift and show that it involves twisted traces of CM values and geodesic cycle integrals of the input function. As an application, we recover Zagier's generating series of twisted traces of singular moduli of weight 1/2 as Millson lifts.

Zagier-lift type arithmetic in harmonic weak Maass forms

Zagier defined lifts of weakly holomorphic modular functions to weakly holomorphic modular forms of weight 3/2. Duke and Jenkins extended Zagier-lifts for weakly holomorphic modular forms of negative-integral weights. In this paper, we show that half-integral weight harmonic weak Maass forms can be interpreted as Zagier-lifts of negative-integral weight harmonic weak Maass forms.

$L$-series for Vector-valued Modular Forms

Motivated by the recent works of Bringmann, Guerzhoy, Kent, and Ono [4] and Bringmann, Fricke, and Kent [3], we introduce $L$-series for vector-valued weakly holomorphic cusp forms, and mock modular period polynomials for vector-valued harmonic weak Maass forms. In particular, we will discuss an integral representation of this new $L$-series and the limiting behavior of special values. Moreover, we also give relations between mock modular periods and $L$-series for vector-valued harmonic weak Maass forms.

Singular invariants and coefficients of harmonic weak Maass forms of weight 5/2

Abstract We study the coefficients of a natural basis for the space of harmonic weak Maass forms of weight 5 / 2 ${5/2}$ on the full modular group. The non-holomorphic part of the first element of this infinite basis encodes the values of the partition function p ⁢ ( n ) ${p(n)}$ . We show that the coefficients of these harmonic Maass forms are given by traces of singular invariants. These are values of non-holomorphic modular functions at CM points or their real quadratic analogues: cycle integrals of such functions along geodesics on the modular curve. The real quadratic case relates to recent work of Duke, Imamoḡlu, and Tóth on cycle integrals of the j-function, while the imaginary quadratic case recovers the algebraic formula of Bruinier and Ono for the partition function.

Bilateral series in terms of mixed mock modular forms

The number of strongly unimodal sequences of weight n is denoted by $u^{*}(n)$ . The generating functions for $\{u^{*}(n)\}_{n=1}^{\infty}$ are $U^{*}(q)=\sum_{n=1}^{\infty}u^{*}(n)q^{n}$ . Rhoades recently gave a precise asymptotic for $u^{*}(n)$ by expressing $U^{*}(q)$ as a mixed mock modular form. In this note, by revisiting the mixed mock modular form associated to $U^{*}(q)$ , three new mixed mock modular forms are constructed by considering the bilateral series of $U^{*}(q)$ and the third order Ramanujan’s mock theta function $f(q)$ . The inner relationships among them are discussed although they are defined in different ways. These new mixed mock modular forms can be expressed in terms of Appell-Lerch sums. The related mock theta functions can be completed as harmonic weak Maass forms. As an application, we give a proof for the claim by Bajpai et al. that the bilateral series $B(f;q)$ of the third order mock theta function $f(q)$ is a mixed mock modular form.

Weakly holomorphic Hecke eigenforms and Hecke eigenpolynomials

In the work of Bringmann, Guerzhoy, Kent and Ono [2] weakly holomorphic Hecke eigenforms are defined and constructed in the level one case by making use of harmonic weak Maass forms. In this paper we extend their results to higher level cases and give an explicit construction in terms of weakly holomorphic modular forms without relying on the theory of harmonic weak Maass forms. Moreover we find a basis for the space of period polynomials consisting of Hecke eigenpolynomials.

Mock modular forms and quantum modular forms

In his last letter to Hardy, Ramanujan introduced mock theta functions. For each of his examples $f(q)$, Ramanujan claimed that there is a collection $\{ G_j\}$ of modular forms such that for each root of unity $\zeta$, there is a $j$ such that \[ \lim _{q \to \zeta }(f(q) - G_j(q)) = O(1).\] Moreover, Ramanujan claimed that this collection must have size larger than $1$. In his 2001 PhD thesis, Zwegers showed that the mock theta functions are the holomorphic parts of harmonic weak Maass forms. In this paper, we prove that there must exist such a collection by establishing a more general result for all holomorphic parts of harmonic Maass forms. This complements the result of Griffin, Ono, and Rolen that shows such a collection cannot have size $1$. These results arise within the context of Zagier’s theory of quantum modular forms. A linear injective map is given from the space of mock modular forms to quantum modular forms. Additionally, we provide expressions for “Ramanujan’s radial limits” as $L$-values.

Linear relations among half-integral weight Poincaré series

We construct an infinite family of half-integral weight Poincaré series coming from vector valued harmonic weak Maass forms, and investigate linear relations among the Poincaré series by applying a pairing between weakly holomorphic modular forms and harmonic weak Maass forms together with properties of Maass Poincaré series. We also show that the Poincaré series are characterized by the Fourier coefficients of half-integral weight cusp forms.