Vector-valued Modular Forms Research Articles

Behavior of theta-series under modular substitutions

In this article we provide a transformation formula of certain theta series, and apply it to obtain non-holomorphic vector-valued modular forms.

Singular Moduli of Shimura Curves

Abstract The j-function acts as a parametrization of the classical modular curve. Its values at complex multiplication (CM) points are called singular moduli and are algebraic integers. A Shimura curve is a generalization of the modular curve and, if the Shimura curve has genus 0, a rational parameterizing function exists and when evaluated at a CM point is again algebraic over Q. This paper shows that the coordinate maps given by N. Elkies for the Shimura curves associated to the quaternion algebras with discriminants 6 and 10 are Borcherds lifts of vector-valued modular forms. This property is then used to explicitly compute the rational norms of singular moduli on these curves. This method not only verifies conjectural values for the rational CM points, but also provides a way of algebraically calculating the norms of CM points with arbitrarily large negative discriminant.

Q-expansions of vector-valued modular forms of negative weight

In this paper, we determine the q-expansions of vector-valued modular forms (Knopp and Mason in Ill. J. Math. 48:1345–1366, 2004; Acta Arith. 110(2): 117–124, 2003) of large negative weight on the full modular group where we allow poles in the upper half plane and at infinity.

Irreducible vector-valued modular forms of dimension less than six

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 forms is a free module over the ring of classical modular forms for the full modular group. A mild restriction is then placed on the class of representation considered in these dimensions, and this again yields an explicit determination of the associated Hilbert-Poincare series.

Logarithmic vector-valued modular forms

We consider logarithmic vector- and matrix-valued modular forms of integral weight $k$ associated with a $p$-dimensional representation $\rho: SL_2(\mathbb{Z}) \to GL_p(\mathbb{C})$ of the modular group, subject only to the condition that $\rho(T)$ has eigenvalues of absolute value 1. The main result is the construction of meromorphic matrix-valued Poincar\'e series associated to $\rho$ 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 $\mathcal{H}(\rho)=\oplus_k \mathcal{H}(k, \rho)$ of all holomorphic logarithmic vector-valued modular forms associated to $\rho$ is a free module of rank $p$ over the ring of classical holomorphic modular forms on $SL_2(\mathbb{Z})$.

Noether-Lefschetz theory and the Yau-Zaslow conjecture

The Yau-Zaslow conjecture predicts the genus 0 curve counts of K 3 K3 surfaces in terms of the Dedekind η \eta function. The classical intersection theory of curves in the moduli of K 3 K3 surfaces with Noether-Lefschetz divisors is related to 3-fold Gromov-Witten invariants via the K 3 K3 curve counts. Results by Borcherds and Kudla-Millson determine these classical intersections in terms of vector-valued modular forms. Proven mirror transformations can often be used to calculate the 3-fold invariants which arise. Via a detailed study of the STU model (determining special curves in the moduli of K 3 K3 surfaces), we prove the Yau-Zaslow conjecture for all curve classes on K 3 K3 surfaces. Two modular form identities are required. The first, the Klemm-Lerche-Mayr identity relating hypergeometric series to modular forms after mirror transformation, is proven here. The second, the Harvey-Moore identity, is proven by D. Zagier and presented in the paper.

Modular differential equations for characters of RCFT

We discuss methods, based on the theory of vector-valued modular forms, to determine all modular differential equations satisfied by the conformal characters of RCFT; these modular equations are related to the null vector relations of the operator algebra. Besides describing effective algorithmic procedures, we illustrate our methods on an explicit example.

Structure of the module of vector-valued modular forms

Let $V$ be a representation of the modular group $\Gamma$ 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 $\Gamma$-modules to graded $\mathcal{M}$-lattices and give some applications, including the calculation of the Hilbert-Poincar\'{e} of $\mathcal{H}(V)$ in some cases.

SYMMETRIC SQUARE L-FUNCTIONS AND SHAFAREVICH–TATE GROUPS, II

We re-examine some critical values of symmetric square L-functions for cusp forms of level one. We construct some more of the elements of large prime order in Shafarevich–Tate groups, demanded by the Bloch–Kato conjecture. For this, we use the Galois interpretation of Kurokawa-style congruences between vector-valued Siegel modular forms of genus two (cusp forms and Klingen–Eisenstein series), making further use of a construction due to Urban. We must assume that certain 4-dimensional Galois representations are symplectic. Our calculations with Fourier expansions use the Eholzer–Ibukiyama generalization of the Rankin–Cohen brackets. We also construct some elements of global torsion which should, according to the Bloch–Kato conjecture, contribute a factor to the denominator of the rightmost critical value of the standard L-function of the Siegel cusp form. Then we prove, under certain conditions, that the factor does occur.

ANALYTIC CONTINUATION OF NONANALYTIC VECTOR-VALUED EISENSTEIN SERIES

In this paper, we introduce a vector-valued nonanalytic Eisenstein series appearing naturally in the Rankin–Selberg convolution of a vector-valued modular cusp form associated to a monomial representation ρ of SL(2,ℤ). This vector-valued Eisenstein series transforms under a representation χρ associated to ρ. We use a method of Selberg to obtain an analytic continuation of this vector-valued nonanalytic Eisenstein series to the whole complex plane.

Bessel Models for Lowest Weight Representations of

We prove uniqueness and give precise criteria for existence of split and nonsplit Bessel models for a class of lowest and highest weight representations of the groups and including all holomorphic and antiholomorphic discrete series representations. Explicit formulas for the resulting Bessel functions are obtained by solving systems of differential equations. The formulas are applied to derive an integral representation for a global L-function on GSp(4) × GL(2) involving a vector-valued Siegel modular form of degree 2.

THE COEFFICIENTS OF THE ω(q) MOCK THETA FUNCTION

In 1920, Ramanujan wrote to Hardy about his discovery of the mock theta functions. In the years since, there has been much work in understanding the transformation properties and asymptotic nature of these functions. Recently, Zwegers proved a relationship between mock theta functions and vector-valued modular forms, and Bringmann and Ono used the theory of Maass forms and Poincaré series to prove a conjecture of Andrews, yielding an exact formula for the coefficients of the f(q) mock theta function. Here we build upon these results, using the theory of vector-valued modular forms and Poincaré series to prove an exact formula for the coefficients of the ω(q) mock theta function.

Siegel Modular Forms of Genus 2 and Level 2: Cohomological Computations and Conjectures

We study the cohomology of certain local systems on moduli spaces of principally polarized abelian surfaces with a level 2 structure. The trace of Frobenius on the alternating sum of the \'etale cohomology groups of these local systems can be calculated by counting the number of pointed curves of genus 2 with a prescribed number of Weierstrass points over the given finite field. This cohomology is intimately related to vector-valued Siegel modular forms. The corresponding scheme in level 1 was carried out in [FvdG]. Here we extend this to level 2 where new phenomena appear. We determine the contribution of the Eisenstein cohomology together with its S_6-action for the full level 2 structure and on the basis of our computations we make precise conjectures on the endoscopic contribution. We also make a prediction about the existence of a vector-valued analogue of the Saito-Kurokawa lift. Assuming these conjectures that are based on ample numerical evidence, we obtain the traces of the Hecke-operators T(p) for p < 41 on the remaining spaces of `genuine' Siegel modular forms. We present a number of examples of 1-dimensional spaces of eigenforms where these traces coincide with the Hecke eigenvalues. We hope that the experts on lifting and on endoscopy will be able to prove our conjectures.

A modern fareytail

We revisit the “fareytail expansions” of elliptic genera which have been used in discussions of the AdS3/CFT2 correspondence and the OSV conjecture. We show how to write such expansions without the use of the problematic “fareytail transform.” In particular, we show how to write a general vector-valued modular form of non-positive weight as a convergent sum over cosets of SL(2,Z). This sum suggests a new regularization of the gravity path integral in AdS3, resolves the puzzles associated with the “fareytail transform,” and leads to several new insights. We discuss constraints on the polar coefficients ofnegative weight modular forms arising from modular invariance, showing how these are related to Fourier coefficients of positive weight cusp forms. In addition, we discuss the appearance of holomorphic anomalies in the context of the fareytail.

Vector-valued Maass-Poincaré series

Shortly before his death, Ramanujan wrote about his discovery of mock theta functions, functions with interesting analytic properties. Recently, Zweger showed that mock theta functions could be completed to satisfy the transformation properties of a weight 1/2 real analytic vector-valued modular form. Using Maass-Poincare series, Bringmann and Ono proved the Andrews-Dragonette conjecture, establishing an exact formula for the coefficients of Ramanujan's mock theta function f(q). In this paper we study vector-valued Maass-Poincare series of all weights, and give their Fourier expansions.

2-Dimensional vector-valued modular forms

We study the graded space ℋ(ρ) of holomorphic vector-valued modular forms of integral weight associated to a 2-dimensional irreducible representation ρ of SL(2,ℤ). When ρ(T) is unitary, a complete description of ℋ(ρ) is given: the Poincaré series is calculated and it is shown that ℋ(ρ) is a free module of rank 2 over the ring of (classical) holomorphic modular forms on SL(2,ℤ).

VECTOR-VALUED MODULAR FORMS AND LINEAR DIFFERENTIAL OPERATORS

We consider holomorphic vector-valued modular forms F of integral weight k on the full modular group Γ = SL(2, ℤ) corresponding to representations of Γ of arbitrary finite dimension p. Assuming that the component functions of F are linearly independent, we prove that the inequality k ≥ 1 - p always holds, and that equality holds only in the trivial case when p = 1 and k = 0. For any p ≥ 2, we show how to construct large numbers of representations of Γ for which k = 2 - p. The key idea is to consider representations of Γ on spaces of solutions of certain linear differential equations whose coefficients are modular forms.

Vector-valued Jacobi-like forms

We introduce vector-valued Jacobi-like forms associated to a representation \(\rho: \Gamma \rightarrow GL(n,{\Bbb C})\) of a discrete subgroup \(\Gamma \subset SL(2,{\Bbb C})\) in \({\Bbb C}^n\) and establish a correspondence between such vector-valued Jacobi-like forms and sequences of vector-valued modular forms of different weights with respect to ρ. We determine a lifting of vector-valued modular forms to vector-valued Jacobi-like forms as well as a lifting of scalar-valued Jacobi-like forms to vector-valued Jacobi-like forms. We also construct Rankin-Cohen brackets for vector-valued modular forms.

Vector-Valued Modular Forms of Weight Two Associated With Jacobi-Like Forms

AbstractWe construct vector-valued modular forms of weight 2 associated to Jacobi-like forms with respect to a symmetric tensor representation of Γ by using the method of Kuga and Shimura as well as the correspondence between Jacobi-like forms and sequences of modular forms. As an application, we obtain vector-valued modular forms determined by theta functions and by pseudodifferential operators.

Cycles with local coefficients for orthogonal groups and vector-valued Siegel modular forms

The purpose of this paper is to generalize the relation between intersection numbers of cycles in locally symmetric spaces of orthogonal type and Fourier coefficients of Siegel modular forms to the case where the cycles have local coefficients. Now the correspondence will involve vector-valued Siegel modular forms.