Holomorphic Modular Forms Research Articles

Effective bounds for traces of singular moduli

We give an asymptotic formula for traces of weak Maass forms at CM points with an effective bound on the error term. Upon specializing to the modular j-function, we deduce such a result for traces of singular moduli. Due to work of Zagier, and Bringmann and Ono, these traces of weak Maass forms at CM points appear as Fourier coefficients of half-integral weight weakly holomorphic modular forms. Hence, our results give effective upper bounds for these Fourier coefficients.

On the Modular Completion of Certain Generating Functions

Abstract We complete several generating functions to non-holomorphic modular forms in two variables. For instance, we consider the generating function of a natural family of meromorphic modular forms of weight two. We then show that this generating series can be completed to a smooth, non-holomorphic modular form of weights $\frac 32$ and two. Moreover, it turns out that the same function is also a modular completion of the generating function of weakly holomorphic modular forms of weight $\frac 32$, which prominently appear in work of Zagier [27] on traces of singular moduli.

Modular forms from the Weierstrass functions

We construct holomorphic elliptic modular forms of weight 2 and weight 1, by special values of Weierstrass p-functions, and by differences of special values of Weierstrass zeta-functions, respectively. Also we calculated the values of these forms at some cusps.

A CLASS OF NONHOLOMORPHIC MODULAR FORMS II: EQUIVARIANT ITERATED EISENSTEIN INTEGRALS

We introduce a new family of real-analytic modular forms on the upper-half plane. They are arguably the simplest class of ‘mixed’ versions of modular forms of level one and are constructed out of real and imaginary parts of iterated integrals of holomorphic Eisenstein series. They form an algebra of functions satisfying many properties analogous to classical holomorphic modular forms. In particular, they admit expansions in $q,\overline{q}$ and $\log |q|$ involving only rational numbers and single-valued multiple zeta values. The first nontrivial functions in this class are real-analytic Eisenstein series.

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.

Constructions of nearly holomorphic Siegel modular forms of degree two

Pitale, Saha and Schmidt studied the representation theoretic aspects of nearly holomorphic modular forms. By their theory, we obtain a classification of [Formula: see text]-modules which occur in the space of nearly holomorphic modular forms. In this paper, we give two constructions of nearly holomorphic Siegel modular forms of degree [Formula: see text] which generate reducible indecomposable modules. One construction is given by the Rankin–Cohen bracket of Shimura’s Eisenstein series and the other by Klingen Eisenstein series.

Quasi‐pullback of Borcherds products

Quasi-pullback of Borcherds products is an operation of renormalized restriction. It produces a meromorphic modular form on a lower dimensional symmetric domain which is again a Borcherds product. We give an explicit formula for the weakly holomorphic modular form of Weil representation type whose Borcherds lift is the quasi-pullback of the given Borcherds product.

Congruences for modular forms and generalized Frobenius partitions

The partition function is known to exhibit beautiful congruences that are often proved using the theory of modular forms. In this paper, we study the extent to which these congruence results apply to the generalized Frobenius partitions defined by Andrews (Mem Am Math Soc 49(301):iv+44, 1984). In particular, we prove that there are infinitely many congruences for $$c\phi _k(n)$$ modulo $$\ell ,$$ where $$\gcd (\ell ,6k)=1,$$ and we also prove results on the parity of $$c\phi _k(n).$$ Along the way, we prove results regarding the parity of coefficients of weakly holomorphic modular forms which generalize work of Ono (J Reine Angew Math 472:1–15, 1996).

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.

On some incomplete theta integrals

In this paper we construct indefinite theta series for lattices of arbitrary signature $(p,q)$ as ‘incomplete’ theta integrals, that is, by integrating the theta forms constructed by the second author with J. Millson over certain singular $q$-chains in the associated symmetric space $D$. These chains typically do not descend to homology classes in arithmetic quotients of $D$, and consequently the theta integrals do not give rise to holomorphic modular forms, but rather to the non-holomorphic completions of certain mock modular forms. In this way we provide a general geometric framework for the indefinite theta series constructed by Zwegers and more recently by Alexandrov, Banerjee, Manschot, and Pioline, Nazaroglu, and Raum. In particular, the coefficients of the mock modular forms are identified with intersection numbers.

Moonshine modules and a question of Griess

We consider the situation in which a finite group acts on an infinite-dimensional graded module in such a way that the graded trace functions are weakly holomorphic modular forms. Under a mild hypothesis we completely describe the asymptotic module structure of the homogeneous subspaces. As a consequence we find that moonshine for a group gives rise to partial orderings on its irreducible representations. This serves as a first answer to a question posed by Griess. In particular, we show that our hypothesis holds for umbral moonshine and for automorphism groups of certain vertex operator algebras.

Congruences involving the $$U_{\ell }$$ operator for weakly holomorphic modular forms

Let $$\lambda $$ be an integer, and $$f(z)=\sum _{n\gg -\infty } a(n)q^n$$ be a weakly holomorphic modular form of weight$$\lambda +\frac{1}{2}$$ on $$\Gamma _0(4)$$ with integral coefficients. Let $$\ell \ge 5$$ 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 $$(f|U_{\ell ^m})(z) = \sum _{n=0}^\infty b(n)q^n$$ has the form $$\begin{aligned} (f|U_{\ell ^m})(z) \equiv b(0) + \sum _{i=1}^{t}\sum _{n=1}^{\infty } b(d_i n^2) q^{d_i n^2} \pmod {\ell }, \end{aligned}$$where $$d_1, \ldots , d_t$$ are square-free positive integers, and the operator $$U_\ell $$ on formal power series is defined by $$\begin{aligned} \left( \sum _{n=0}^\infty a(n)q^n \right) \bigg | U_\ell = \sum _{n=0}^\infty a(\ell n)q^n. \end{aligned}$$Then, $$\lambda \equiv 0 \pmod {\frac{\ell -1}{2}}$$. Moreover, if $${\tilde{f}}$$ denotes the coefficient-wise reduction of f modulo $$\ell $$, then we have $$\begin{aligned} \biggl \{ \lim _{m \rightarrow \infty } {\tilde{f}}|U_{\ell ^{2m}}, \lim _{m \rightarrow \infty } {\tilde{f}}|U_{\ell ^{2m+1}} \biggr \} = \biggl \{ a(0)\theta (z), a(0)\theta ^\ell (z) \in \mathbb {F}_{\ell }[[q]] \biggr \}, \end{aligned}$$where $$\theta (z)$$ is the Jacobi theta function defined by $$\theta (z) = \sum _{n\in \mathbb {Z}} q^{n^2}$$. By using this result, we obtain the distribution of the Fourier coefficients of weakly holomorphic modular forms in congruence classes. This applies to the congruence properties for traces of singular moduli.

Honda–Kaneko congruences and the Mazur–Tate p-adic σ-function

Several years ago, Honda and Kaneko experimentally discovered remarkable congruences for Fourier coefficients of a certain weakly holomorphic modular form. The congruences attracted interest, and have been proved since then using various approaches. In this paper, we suggest yet another approach to these congruences. Specifically, the very existence of the p-adic σ-function constructed by Mazur and Tate allows us to prove and expand the congruences.

Differential operators on polar harmonic Maass forms and elliptic duality

Abstract In this paper, we study polar harmonic Maass forms of negative integral weight. Using work of Fay, we construct Poincaré series which span the space of such forms and show that their elliptic coefficients exhibit duality properties which are similar to the properties known for Fourier coefficients of harmonic Maass forms and weakly holomorphic modular forms.

Newforms of half-integral weight: the minus space of Sk+1/2(Γ0(8M))

We compute generators and relations for a certain 2-adic Hecke algebra of level 8 associated with the double cover of SL2 and a 2-adic Hecke algebra of level 4 associated with PGL2. We show that these two Hecke algebras are isomorphic as expected from the Shimura correspondence. We use the 2-adic generators to define classical Hecke operators on the space of holomorphic modular forms of weight k + 1/2 and level 8M where M is odd and square-free. Using these operators and our previous results on half-integral weight forms of level 4M we define a subspace of the space of half-integral weight forms as a common -1 eigenspace of certain Hecke operators. Using the relations and a result of Ueda we show that this subspace, which we call the minus space, is isomorphic as a Hecke module under the Ueda correspondence to the space of new forms of weight 2k and level 4M. We observe that the forms in the minus space satisfy a Fourier coefficient condition that gives the complement of the plus space but does not define the minus space.

Rationality and p-adic properties of reduced forms of half-integral weight

In this paper we study special bases of certain spaces of half-integral weight weakly holomorphic modular forms. We establish a criterion for the integrality of Fourier coefficients of such bases. By using recursive relations between Hecke operators, we derive relations of Fourier coefficients of each basis element and obtain congruences of the Fourier coefficients, which extend known congruences for traces of singular moduli.

Modular cocycles and cup product

The Eichler-Shimura isomorphism relates holomorphic modular cusps forms of positive even integral weight to cohomology classes. The Haberland formula uses the cup product to give a cohomological formulation of the Petersson scalar product.In this paper we extend Haberland's formula to modular cusp forms of positive real weight. This relation is based on the cup product of an Eichler cocycle and a Knopp cocycle.We may also consider the cup product of two Eichler cocycles. In the classical situation this cup product is almost always zero. However we show evidence that for real weights this cup product may very well be non-trivial. We approach the question whether the cup product is a non-trivial coinvariant by duality with a space of entire modular forms. The cup product yields a bilinear map over C from pairs of holomorphic modular forms (not necessarily of the same weight, one of them may have large growth at the cusps) to coinvariants in infinite-dimensional modules. To investigate whether this bilinear map is non-trivial we test the result against entire modular forms of a suitable weight. Under some conditions on the weights, this leads to an explicit triple integral, which can be investigated numerically, thus providing evidence that the cup product is non-trivial at least in some situations.

The transcendence of zeros of canonical basis elements of the space of weakly holomorphic modular forms for Γ0(2)

We consider the canonical basis elements fk,mε for the space of weakly holomorphic modular forms of weight k for the Hecke congruence group Γ0(2) and we prove that for all m≥c(k) for some constant c(k), if z0 in a fundamental domain for Γ0(2) is a zero of fk,mε, then either z0 is in {i2,−12+i2,12+i2,−1+i74,1+i74}, or z0 is transcendental.

A basis for the space of weakly holomorphic Drinfeld modular forms for GL2(A)

We construct a canonical basis for the space of weakly holomorphic Drinfeld modular forms. And we find that the basis elements satisfy a generating function and the duality among coefficients of the basis elements. Moreover we obtain the congruence properties of t-expansion coefficients of these functions under some conditions.

Modularity of Galois traces of Weber's resolvents

The values of the classical j-invariant at CM points are called singular moduli. Zagier [15] proved that the traces of singular moduli are Fourier coefficients of a weakly holomorphic modular form of weight 3/2 and Bruinier-Funke [1] generalized his result to the sums of the values at Heegner points of modular functions on modular curves of arbitrary genus. Extending the work of [9], we construct the real-valued class invariants by using the modular functions defined by Weber's resolvents and identify their Galois traces with Fourier coefficients of a weight 3/2 weakly holomorphic modular form by using Shimura's reciprocity law.