Holomorphic Modular Forms Of Weight Research Articles

Algebra of Borcherds products

Borcherds lift for an even lattice of signature ( p , q ) (p, q) is a lifting from weakly holomorphic modular forms of weight ( p − q ) / 2 (p-q)/2 for the Weil representation. We introduce a new product operation on the space of such modular forms and develop a basic theory. The product makes this space a finitely generated filtered associative algebra, without unit element and noncommutative in general. This is functorial with respect to embedding of lattices by the quasi-pullback. Moreover, the rational space of modular forms with rational principal part is closed under this product. In some examples with p = 2 p=2 , the multiplicative group of Borcherds products of integral weight forms a subring.

On the zeros of certain weakly holomorphic modular forms for ${\varGamma }_0^+(5)$ and ${\varGamma }_0^+(7)$

Let $p$ be a prime for which $\Gamma _0^+(p)$ is of genus zero, and let $M_k^!(\Gamma _0^+(p))$ be the space of weakly holomorphic modular forms of weight $k$ for $\Gamma _0^+(p) $. It is known that $M_k^!(\Gamma _0^+(p))$ has the natural basis. In this p

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.

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.

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.

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.

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.

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.

Arithmetic properties for the minus space of weakly holomorphic modular forms

Let Mk!(p) be the space of weakly holomorphic modular forms of weight k on Γ0(p), and let Mk!−(p) be the minus space which is the subspace of Mk!(p) consisting of all eigenforms of the Fricke involution Wp with eigenvalue −1. We are interested in finding a canonical basis for the minus space Mk!−(p) for certain levels. Using this result, along with previous works of Choi and Kim [CK13], we find a canonical basis for the space Mk!(p), and investigate its arithmetic properties. We also give another generalization of [CK13] to the cases of square-free integer levels.

Hecke structures of weakly holomorphic modular forms and their algebraic properties

Let Sk!(Γ1(N)) be the space of weakly holomorphic cusp forms of weight k on Γ1(N) with an even integer k>2 and Mk!(Γ1(N)) be the space of weakly holomorphic modular forms of weight k on Γ1(N). Further, let z denote a complex variable and D:=12πi∂∂z. In this paper, we construct a basis of the space Sk!(Γ1(N))/Dk−1(M2−k!(Γ1(N))) consisting of Hecke eigenforms by using the Eichler–Shimura cohomology theory. Further, we study algebraicity of CM values of weakly holomorphic modular forms in the basis. This applies to an analogue of the Chowla–Selberg formula for a mock modular form whose shadow is the Ramanujan delta function.

Regularized Inner Products and Errors of Modularity

We develop a regularization for Petersson inner products of arbitrary weakly holomorphic modular forms, generalizing several known regularizations. As one application, we extend work of Duke, Imamoglu, and Toth on regularized inner products of weakly holomorphic modular forms of weights 0 and 3=2. These regularized inner products can be evaluated in terms of the coefficients of holomorphic parts of harmonic Maass forms of dual weights. Moreover, we study the errors of modularity of the holomorphic parts of such a harmonic Maass forms and show that they induce cocyles in the first parabolic cohomology group introduced by Bruggeman, Choie, and the second author. This provides explicit representatives of the cohomology classes constructed abstractly and in a very general setting in their work.

Interlacing of zeros of certain weakly holomorphic modular forms for [formula omitted

We prove that zeros of each basis element of the space of weakly holomorphic modular forms of weight k for the Fricke group Γ0+(2) of level 2 interlace, extending the result for SL2(Z) of Jenkins and Pratt [4].

A proof of the Thompson moonshine conjecture

In this paper we prove the existence of an infinite dimensional graded super-module for the finite sporadic Thompson group $Th$ whose McKay-Thompson series are weakly holomorphic modular forms of weight $\frac 12$ satisfying properties conjectured by Harvey and Rayhaun.

Zeros of weakly holomorphic modular forms of level 5

Let [Formula: see text] be the space of weakly holomorphic modular forms of weight [Formula: see text] and level [Formula: see text] that are holomorphic away from the cusp at [Formula: see text]. We study a canonical basis for [Formula: see text] and the locations of zeros of this basis in a fundamental domain. We give a lower bound for the number of zeros lying on a lower boundary arc of the fundamental domain.

Heegner divisors in generalized Jacobians and traces of singular moduli

We prove an abstract modularity result for classes of Heegner divisors in the generalized Jacobian of a modular curve associated to a cuspidal modulus. Extending the Gross-Kohnen-Zagier theorem, we prove that the generating series of these classes is a weakly holomorphic modular form of weight 3/2. Moreover, we show that any harmonic Maass forms of weight 0 defines a functional on the generalized Jacobian. Combining these results, we obtain a unifying framework and new proofs for the Gross-Kohnen-Zagier theorem and Zagier's modularity of traces of singular moduli, together with new geometric interpretations of the traces with non-positive index.

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.

Weak harmonic Maass forms of weight 5/2 and a mock modular form for the partition function

We construct a natural basis for the space of weak harmonic Maass forms of weight 5/2 on the full modular group. The nonholomorphic part of the first element of this basis encodes the values of the ordinary partition function p(n). We obtain a formula for the coefficients of the mock modular forms of weight 5/2 in terms of regularized inner products of weakly holomorphic modular forms of weight −1/2, and we obtain Hecke-type relations among these mock modular forms.

A Gross–Kohnen–Zagier type theorem for higher-codimensional Heegner cycles

We prove that Heegner cycles of codimension m+1 inside Kuga-Sato type varieties of dimension 2m+1 are coefficients of modular forms of weight 3/2+m in the appropriate quotient group. The main technical tool for generating the necessary relations is a Borcherds style theta lift with polynomials. We also show how this lift defines a new singular Shimura-type correspondence from weakly holomorphic modular forms of weight 1/2-m to meromorphic modular forms of weight 2m+2.

L ∞ norms of holomorphic modular forms in the case of compact quotient

Abstract We prove a sub-convex estimate for the sup-norm of L 2-normalized holomorphic modular forms of weight k on the upper half plane, with respect to the unit group of a quaternion division algebra over ℚ. More precisely we show that when the L 2 norm of an eigenfunction f is one, ∥ f ∥ ∞ ≪ ε k 1 2 - 1 33 + ε $ \Vert f \Vert _\infty \ll _\varepsilon k^{\frac{1}{2} - \frac{1}{33} +\varepsilon } $ for any ε > 0 ${\varepsilon >0}$ and for all k sufficiently large.

Periods of the j -function along infinite geodesics and mock modular forms

Zagier's well-known work on traces of singular moduli relates the coefficients of certain weakly holomorphic modular forms of weight 1 2 to traces of values of the modular j-function at imaginary quadratic points. A real quadratic analogue was recently studied by Duke, Imamoḡlu, and Tóth. They showed that the coefficients of certain weight 1 2 mock modular forms f D = ∑ d > 0 a ( d , D ) q d , D > 0 are given in terms of traces of cycle integrals of the j-function. Their result applies to those coefficients a ( d , D ) for which d D is not a square. Recently, Bruinier, Funke, and Imamoḡlu employed a regularized theta lift to show that the coefficients a ( d , D ) for square d D are traces of regularized integrals of the j-function. In the present paper, we provide an alternate approach to this problem. We introduce functions j m , Q (for Q a quadratic form) which are related to the j-function and show, by modifying the method of Duke, Imamoḡlu, and Tóth, that the coefficients for which d D is a square are traces of cycle integrals of the functions j m , Q .