Let $V$ be a representation of the modular group $Γ$ of dimension $p$. We show that the $\mathbb{Z}$-graded space $\mathcal{H}(V)$ of holomorphic vector-valued modular forms associated to $V$ is a free module of rank $p$ over the algebra $\mathcal{M}$ of classical holomorphic modular forms. We study the nature of $\mathcal{H}$ considered as a functor from $Γ$-modules to graded $\mathcal{M}$-lattices and give some applications, including the calculation of the Hilbert-Poincaré of $\mathcal{H}(V)$ in some cases.
We extend the relation between quasi-modular forms and modular forms to a wider class of functions. We then relate both forms to vector-valued modular forms with symmetric power representations, and prove a general structure theorem for these vector-valued forms.
Let $H$ denote a finite index subgroup of the modular group $Γ$ and let $ρ$ denote a finite-dimensional complex representation of $H.$ Let $M(ρ)$ denote the collection of holomorphic vector-valued modular forms for $ρ$ and let $M(H)$ denote the collection of modular forms on $H$. Then $M(ρ)$ is a $\textbf{Z}$-graded $M(H)$-module. It has been proven that $M(ρ)$ may not be projective as a $M(H)$-module. We prove that $M(ρ)$ is Cohen-Macaulay as a $M(H)$-module. We also explain how to apply this result to prove that if $M(H)$ is a polynomial ring then $M(ρ)$ is a free $M(H)$-module of rank $\tex…
We survey the theory of vector-valued modular forms and their connections with modular differential equations and Fuchsian equations over the three-punctured sphere. We present a number of numerical examples showing how the theory in dimensions 2 and 3 leads naturally to close connections between modular forms and hypergeometric series.
We classify the three-dimensional representations of the modular group that are reducible but indecomposable, and their associated spaces of holomorphic vector-valued modular forms. We then demonstrate how such representations may be employed to compute periods of modular curves. This technique obviates the use of Hecke operators, and therefore provides a method for studying noncongruence modular curves as well as congruence.
This paper gives a simple method for constructing vector-valued Siegel modular forms from scalar-valued ones. The method is efficient in producing the siblings of Delta, the smallest weight cusp forms that appear in low degrees. It also shows the strong relations between these modular forms of different genera. We illustrate this by a number of examples.
We characterize all logarithmic, holomorphic vector-valued modular forms which can be analytically continued to a region strictly larger than the upper half-plane.
In this note, we consider discriminant forms that are given by the norm form of real quadratic fields and their induced Weil representations. We prove that there exists an isomorphism between the space of vector-valued modular forms for the Weil representations that are invariant under the action of the automorphism group and the space of scalar-valued modular forms that satisfy some epsilon-condition, with which we translate Borcherds's theorem of obstructions to scalar-valued modular forms. In the end, we consider an example in the case of level 12.
We define Hecke operators on vector valued modular forms transforming with the Weil representation associated to a discriminant form. We describe the properties of the corresponding algebra of Hecke operators and study the action on modular forms.
We prove the following theorem. Suppose that $F=(f_1, f_2)$ is a 2-dimensional vector-valued modular form on $SL_2(Z)$ whose component functions $f_1, f_2$ have rational Fourier coefficients with bounded denominators. Then $f_1$ and $f_2$ are classical modular forms on a congruence subgroup of the modular group.
We present a formula for vector-valued modular forms, expressing the value of the Hilbert-polynomial of the module of holomorphic forms evaluated at specific arguments in terms of traces of representation matrices, restricting the weight distribution of the free generators.
We give a geometrical construction of the canonical automorphic factor for the Jacobi group and construct new vector valued modular forms from Jacobi forms by differentiating them with respect to toroidal variables and then evaluating at zero.
We construct isomorphisms between spaces of vector-valued modular forms for the dual Weil representation and certain spaces of scalar-valued modular forms in the case that the underlying finite quadratic module $A$ has order $p$ or $2p$, where $p$ is an odd prime. The isomorphisms are given by twisted sums of the components of vector-valued modular forms. Our results generalize work of Bruinier and Bundschuh to the case that the components $F_γ$ of the vector-valued modular form are antisymmetric in the sense that $F_γ = -F_{-γ}$ for all $γ\in A$. As an application, we compute restrictions of …
The theory of elliptic modular forms has gained significant momentum from the discovery of relaxed yet well-behaved notions of modularity, such as mock modular forms, higher order modular forms, and iterated Eichler-Shimura integrals. Applications beyond number theory range from combinatorics, geometry, and representation theory to string theory and conformal field theory. We unify these relaxed notions in the framework of vector-valued modular forms by introducing a new class of $\mathrm{SL}_{2}(\mathbb{Z})$-representations: virtually real-arithmetic types. The key point of the paper is that …
This paper presents the theory of holomorphic vector valued modular forms from a geometric perspective. More precisely, we define certain holomorphic vector bundles on the modular orbifold of generalized elliptic curves whose sections are vector valued modular forms. This perspective simplifies the theory, and it clarifies the role that exponents of representations of SL_2(Z) play in the holomorphic theory of vector valued modular forms. Further, it allows one to use standard techniques in algebraic geometry to deduce free-module theorems and dimension formulae (deduced previously by other aut…
We investigate the various types of weight raising and weight lowering operators on quasi-modular forms, or equivalently on Shimura's vector-valued modular forms involving symmetric power representations. We also present all the eigenfunctions of the two possible Laplacian operators.
We present a dimension formula for spaces of vector-valued modular forms of integer weight in case the associated multiplier system has finite image, and discuss the weight distribution of the module generators of holomorphic and cusp forms, as well as the duality relation between cusp forms and holomorphic forms for the contragredient.
Let $ρ: SL(2,\mathbb{Z})\to GL(2,\mathbb{C})$ be an irreducible representation of the modular group such that $ρ(T)$ has finite order $N$. We study holomorphic vector-valued modular forms $F(τ)$ of integral weight associated to $ρ$ which have \emph{rational} Fourier coefficients. (These span the complex space of all integral weight vector-valued modular forms associated to $ρ$.) As a special case of the main Theorem, we prove that if $N$ does \emph{not} divide 120 then every nonzero $F(τ)$ has Fourier coefficients with \emph{unbounded denominators}.
We carry out some computations of vector valued Siegel modular forms of degree two, weight (k,2) and level one. Our approach is based on Satoh's description of the module of vector-valued Siegel modular forms of weight (k, 2) and an explicit description of the Hecke action on Fourier expansions. We highlight three experimental results: (1) we identify a rational eigenform in a three dimensional space of cusp forms, (2) we observe that non-cuspidal eigenforms of level one are not always rational and (3) we verify a number of cases of conjectures about congruences between classical modular forms…
We define graded hyper-algebras of vector-valued Siegel modular forms, which allow us to study tensor products of the latter. We also define vector-valued Hecke operators for Siegel modular forms at all places of ${\mathbb Q}$, acting on these hyper-algebras. These definitions bridge the classical and representation theoretic approach to Siegel modular forms. Combining both the product structure and the action of Hecke operators, we prove in the case of elliptic modular forms that all cusp forms of sufficiently large weight can be obtained from products involving only two fixed Eisenstein seri…
Rankin-Cohen Bracket for Vector-Valued Modular Forms
Korea Institute for Advanced Study（韩国高等科学研究院）；Sungkyunkwan University（成均馆大学）；American University of Beirut (AUB)（贝鲁特美国大学）；Center for Advanced Mathematical Sciences (CAMS) at AUB（贝鲁特美国大学高级数学科学中心）
We consider logarithmic vector- and matrix-valued modular forms of integral weight $k$ associated with a $p$-dimensional representation $ρ: SL_2(\mathbb{Z}) \to GL_p(\mathbb{C})$ of the modular group, subject only to the condition that $ρ(T)$ has eigenvalues of absolute value 1. The main result is the construction of meromorphic matrix-valued Poincaré series associated to $ρ$ for all large enough weights. The component functions are logarithmic $q$-series, i.e., finite sums of products of $q$-series and powers of $\log q$. We derive several consequences, in particular we show that the space $\…
A general theory of vector-valued modular functions, holomorphic in the upper half-plane, is presented for finite dimensional representations of the modular group. This also provides a description of vector-valued modular forms of arbitrary half-integer weight. It is shown that the space of these modular functions is spanned, as a module over the polynomials in J, by the columns of a matrix that satisfies an abstract hypergeometric equation, providing a simple solution of the Riemann-Hilbert problem for representations of the modular group. Restrictions on the coefficients of this differential…
A thorough analysis is made of the Fourier coefficients for vector-valued modular forms associated to three-dimensional irreducible representations of the modular group. In particular, the following statement is verified for all but a finite number of equivalence classes: if a vector-valued modular form associated to such a representation has rational Fourier coefficients, then these coefficients have "unbounded denominators", i.e. there is a prime number p, depending on the representation, which occurs to an arbitrarily high power in the denominators of the coefficients. This provides a verif…
Let $E_n$ be the Siegel Eisenstein series of degree $n$ and weight $k$ with a complex parameter $s$. In this paper, using a differential operator $D$ by Ibukiyama which sends a scalar valued Siegel modular form to the tensor product of two vector valued Siegel modular forms, under a certain condition, we give a formula of $DE_{p+q}$ on $H_p\times H_q$, where $H_n$ is the Siegel upper half space of degree $n$. Furthermore, we give some applications of this formula, i.e., analytic properies of standard $L$-functions and the Klingen Eisenstein series and algebraicity results for Siegel modular fo…
Cléry and van der Geer determined generators for some modules of vector valued Picard modular forms on the two dimensional ball. In this paper we consider the case of a three dimensional ball with the action of the Picard modular group $Γ_3[\sqrt{-3}]$. The corresponding modular variety of dimension 3 is a copy of the Segre cubic.
We introduce and investigate an infinite family of functions which are shown to have generalised quantum modular properties. We realise their "companions" in the lower half plane both as double Eichler integrals and as non-holomorphic theta functions with coefficients given by double error functions. Further, we view these Eichler integrals in a modular setting as parts of certain weight two indefinite theta series.
In this paper we will describe all vector-valued Siegel modular forms of degree 2 and weight ${\rm Sym}^6({\rm St}) \otimes \det^{k}({\rm St})$ with $k$ odd. These vector-valued forms constitute a module over the ring of classical Siegel modular forms of degree 2 and even weight and this module turns out to be free. In order to find generators, we generalize certain Rankin-Cohen differential operators on triples of classical Siegel modular forms that were first considered by Ibukiyama and we find a Rankin-Cohen bracket on vector-valued Siegel modular forms.
Extending the method of the paper [FS3] we prove three structure theorems for vector valued modular forms, where two correspond to 4-dimensional cases (two hermitian modular groups, one belonging to the field of Eisenstein numbers, the other to the field of Gaussian numbers.) and one to a 6-dimensional case (a quaternionic modular group).
Since their definition in 2010 by Zagier, quantum modular forms have been connected to numerous different topics such as strongly unimodal sequences, ranks, cranks, and asymptotics for mock theta functions near roots of unity. These are functions that are not necessarily defined on the upper half plane but a priori are defined only on a subset of $\Q$, and whose obstruction to modularity is some analytically "nice" function. Motivated by Zagier's example of the quantum modularity of Kontsevich's "strange" function $F(q)$, we revisit work of Andrews, Jimenez-Urroz, and Ono to construct a natura…
In this work we use the Rankin-Selberg method to obtain results on the analytic properties of the standard $L$-function attached to vector valued Siegel modular forms. In particular we provide a detailed description of its possible poles and obtain a non-vanishing result of the twisted $L$-function beyond the usual range of absolute convergence. We remark that these results were known in this generality only in the case of scalar weight Siegel modular forms. As an interesting by-product of our work we establish the cuspidality of some theta series.
We extend Igusa's description of the relation between invariants of binary sextics and Siegel modular forms of degree two to a relation between covariants and vector-valued Siegel modular forms of degree two. We show how this relation can be used to effectively calculate the Fourier expansions of Siegel modular forms of degree two.
In this paper, we prove vector-valued higher depth quantum modular properties arising from characters of certain vertex algebras. We then find two-dimensional Mordell integral representations for their errors of modularity.
We construct generators for modules of vector-valued Picard modular forms on a unitary group of type (2,1) over the Eisenstein integers. We also calculate eigenvalues of Hecke operators acting on cusp forms.
We prove that the field generated by the Fourier coefficients of weakly holomorphic Poincaré series of a given level $Γ_0(N)$ and weight $k\ge 2$ coincides with the field generated by the single-valued periods of a certain motive attached to $Γ_0(N)$. This clarifies the arithmetic nature of such Fourier coefficients and generalises previous formulas of Brown and Acres--Broadhurst giving explicit series expansions for the single-valued periods of some modular forms. Our proof is based on Bringmann--Ono's construction of harmonic lifts of Poincaré series.
An algebraic classification is given for spaces of holomorphic vector-valued modular forms of arbitrary real weight and multiplier system, associated to irreducible, T-unitarizable representations of the full modular group, of dimension less than six. For representations of dimension less than four, it is shown that the associated space of vector-valued modular forms is a cyclic module over a certain skew polynomial ring of differential operators. For dimensions four and five, a complete list of possible Hilbert-Poincare series is given, using the fact that the space of vector-valued modular f…
Let $ρ$ denote an irreducible two-dimensional representation of $Γ_{0}(2)$. The collection of vector-valued modular forms for $ρ$, which we denote by $M(ρ)$, form a graded and free module of rank two over the ring of modular forms on $Γ_{0}(2)$, which we denote by $M(Γ_{0}(2))$. For a certain class of $ρ$, we prove that if Z is any vector-valued modular form for $ρ$ whose component functions have algebraic Fourier coefficients then the sequence of the denominators of the Fourier coefficients of both component functions of Z is unbounded. Our methods involve computing an explicit basis for $M(ρ…
The first half of this dissertation reviews the basic notion of vector-valued modular forms and its connection to differential equations. The main purpose of the dissertation is to classify spaces of vector-valued modular forms associated to irreducible, T-unitarizable representations of the full modular group, of dimension less than six. Given such a representation, it is shown that the associated graded complex linear space of vector-valued modular forms is a free module over the ring of integral weight modular forms for the full modular group, whose rank is equal to the dimension of the giv…
We study Hecke operators on vector-valued modular forms for the Weil representation $ρ_L$ of a lattice $L$. We first construct Hecke operators $\mathcal{T}_r$ that map vector-valued modular forms of type $ρ_L$ into vector-valued modular forms of type $ρ_{L(r)}$, where $L(r)$ is the lattice $L$ with rescaled bilinear form $(\cdot, \cdot)_r = r (\cdot, \cdot)$, by lifting standard Hecke operators for scalar-valued modular forms using Siegel theta functions. The components of the vector-valued Hecke operators $\mathcal{T}_r$ have appeared in [Comm. Math. Phys. 350 (2017), 1069-1121] as generating…
We study vector-valued Siegel modular forms of genus 2 and level 2. We describe the structure of certain modules of vector-valued modular forms over rings of scalar-valued modular forms.
Using the language of vertex operator algebras (VOAs) and vector-valued modular forms we study the modular group representations and spaces of 1-point functions associated to intertwining operators for Virasoro minimal model VOAs. We examine all representations of dimension less than four associated to irreducible modules for minimal models, and determine when the kernel of these representations is a congruence or noncongruence subgroup of the modular group. Arithmetic criteria are given on the indexing of the irreducible modules for minimal models that imply the associated modular group repre…
For any discrete group $Γ$ and any 2-dimensional complex representation $ρ$ of $Γ$, we introduce the notion of $ρ-$equivariant functions, and we show that they are parameterized by vector-valued modular forms. We also provide examples arising from the monodromy of differential equations.
We develop two structure theorems for vector valued Siegel modular forms for Igusa's subgroup Γ_2[2,4], the multiplier system induced by the theta constants and the representation Sym^2. In the proof, we identify some of these modular forms with rational tensors with easily handleable poles on P^3\C. It follows that the observed modules of modular forms are generated by the Rankin-Cohen brackets of the four theta series of the second kind.
We correct a mistake in \cite{St} leading to erroneous formulas in Theorems 5.2 and 5.4. As an immediate corollary of a formula in \cite{BCJ} we give a formula, which relates the Hecke operators $T(p^2)\circ T(p^{2l-2})$, $T(p^{2l})$ and $T(p^{2l-4})$ and comment on it.
We formulate a conjecture that describes the vector-valued Siegel modular forms of degree 2 and level 2 of weight Sym^j det^2 and provide some evidence for it. We construct such modular forms of weight (j,2) via covariants of binary sextics and calculate their Fourier expansions illustrating the effectivity of the approach via covariants. Two appendices contain related results of Chenevier; in particular a proof of the fact that every modular form of degree 2 and level 2 and weight (j,1) vanishes.
The theta correspondence has been an important tool in studying cycles in locally symmetric spaces of orthogonal type. We generalize the Kudla-Millson relation between intersection numbers of cycles and Fourier coefficients of Siegel modular forms to the case where the cycles have local coefficients. Now the generating series of the cycles give rise to vector-valued Siegel modular forms. The underlying correspondence between the highest weights of the orthogonal and the symplectic group coincides with the one obtained by Adams for which we provide a geometric interpretation.
We proof a Garrett-Böcherer decomposition of a vector valued Siegel Eisenstein series $E_{l,0}^2$ of genus 2 transforming with the Weil representation of $\text{Sp}_2(\mathbb{Z})$ on the group ring $\mathbb{C}[(L'/L)^2]$. We show that the standard zeta function associated to a vector valued common eigenform $f$ for the Weil representation can be meromorphically continued to the whole $s$-plane and that it satisfies a functional equation. The proof is based on an integral representation of this zeta function in terms of $f$ and $E_{l,0}^2$.
The mock theta conjectures are ten identities involving Ramanujan's fifth-order mock theta functions. The conjectures were proven by Hickerson in 1988 using q-series methods. Using methods from the theory of harmonic Maass forms, specifically work of Zwegers and Bringmann-Ono, Folsom reduced the proof of the mock theta conjectures to a finite computation. Both of these approaches involve proving the identities individually, relying on work of Andrews-Garvan. Here we give a unified proof of the mock theta conjectures by realizing them as an equality between two nonholomorphic vector-valued modu…
In this article we show that 'most' of the vector valued modular forms w.r.t. the Weil representation on the groups rings $\mathbb{C}[D]$ of discriminant forms D are oldforms. The precise meaning of oldform is that the form can be represented as a sum of lifts of vector valued modular forms on group rings of quotients $H^\bot/H$ for isotropic subgroups H of D. In this context, 'most' means that all forms are oldforms if there is a part $\mathbb{Z}/p^e \mathbb{Z}$ inside a p-part of D that is repeated several times (i.e. >= 4,5,7,9 depending on p and e). We will proceed by giving an oldform det…
We establish a correspondence between vector-valued modular forms with respect to a symmetric tensor representation and quasimodular forms. This is carried out by first obtaining an explicit isomorphism between the space of vector-valued modular forms with respect to a symmetric tensor representation and the space of finite sequences of modular forms of certain type. This isomorphism uses Rankin-Cohen brackets and extends a result of Kuga and Shimura, who considered the case of vector-valued modular forms of weight two. We also obtain a correspondence between such vector-valued modular forms a…
Let $F$ be a nearly holomorphic vector-valued Siegel modular form of weight $ρ$ with respect to some congruence subgroup of $\mathrm{Sp}_{2n}(\mathbb Q)$. In this note, we prove that the function on $\mathrm{Sp}_{2n}(\mathbb R)$ obtained by lifting $F$ has the moderate growth (or "slowly increasing") property. This is a consequence of the following bound that we prove: $\|ρ(Y^{1/2})F(Z) \| \ll \prod_{i=1}^n (μ_i(Y)^{λ_1/2} + μ_i(Y)^{-λ_1/2})$ where $ λ_1 \ge \ldots \ge λ_n$ is the highest weight of $ρ$ and $μ_i(Y)$ are the eigenvalues of the matrix $Y$.
We establish an isomorphism between certain complex-valued and vector-valued modular form spaces of half-integral weight, generalizing the well-known isomorphism between modular forms for $Γ_0(4)$ with Kohnen's plus condition and modular forms for the Weil representation associated to the discriminant form for the lattice with Gram matrix $(2)$. With such an isomorphism, we prove the Zagier duality and write down the Borcherds lifts explicitly.
