The Lie theory of the Rankin-Cohen brackets and allied bi-differential operators

The Lie theoretic nature of the Rankin-Cohen brackets is here uncovered. These bilinear operations, which, among other purposes, were devised to produce a holomorphic automorphic form from any pair of such forms, are instances of SL(2,R)-equivariant holomorphic bi-differential operators on the upper half-plane. All of the latter are here characterized and explicitly obtained, by establishing their one-to-one correspondence with singular vectors in the tensor product of two sl(2,C) Verma modules. The Rankin-Cohen brackets arise in the generic situation where the linear span of the singular vectors of a given weight is one-dimensional. The picture is completed by the special brackets which appear for the finite number of pairs of initial lowest weights for which the above space is two-dimensional. Explicit formulæ for basis vectors in both situations are obtained and universal Lie algebraic objects subsuming all of them are exhibited. A few applications of these results and Lie theoretic approach are then considered. First, a generalization of the latter yields Rankin-Cohen type brackets for Hilbert modular forms. Then, some Rankin-Cohen brackets are shown to intertwine the tensor product of two holomorphic discrete series representations of SL(2,R) with another such representation occurring in the tensor product decomposition. Finally, the sought for precise relationship between the Rankin-Cohen brackets and Gordan's transvection processes of the nineteenth century invariant theory is unveiled.     

Rankin-Cohen brackets on pseudodifferential operators

Pseudodifferential operators that are invariant under the action of a discrete subgroup Γ of SL(2,R) correspond to certain sequences of modular forms for Γ. Rankin-Cohen brackets are noncommutative products of modular forms expressed in terms of derivatives of modular forms. We introduce an analog of the heat operator on the space of pseudodifferential operators and use this to construct bilinear operators on that space which may be considered as Rankin-Cohen brackets. We also discuss generalized Rankin-Cohen brackets on modular forms and use these to construct certain types of modular forms.     

Combinatorics of Maass-Shimura operators

Maass-Shimura operators on holomorphic modular forms preserve the modularity of modular forms but not holomorphy, whereas the derivative preserves holomorphy but not modularity. Rankin-Cohen brackets are bilinear operators that preserve both and are expressed in terms of the derivatives of modular forms. We give identities relating Maass-Shimura operators and Rankin-Cohen brackets on modular forms and obtain a natural expression of the Rankin-Cohen brackets in terms of Maass-Shimura operators. We also give applications to values of L-functions and Fourier coefficients of modular forms.     

Hilbert modular pseudodifferential operators

We introduce Jacobi-like forms of several variables, and study their connections with Hilbert modular forms and pseudodifferential operators of several variables. We also construct Rankin-Cohen brackets for Hilbert modular forms using such Jacobi-like forms.

     

Rankin-Cohen brackets and formal quantization

In this paper, we use the theory of deformation quantization to understand Connes' and Moscovici's results [A. Connes, H. Moscovici, Rankin-Cohen brackets and the Hopf algebra of transverse geometry, Mosc. Math. J. 4 (1) (2004) 111-130, 311]. We use Fedosov's method of deformation quantization of symplectic manifolds to reconstruct Zagier's deformation [D. Zagier, Modular forms and differential operators, in: K.G. Ramanathan Memorial Issue, Proc. Indian Acad. Sci. Math. Sci. 104 (1) (1994) 57-75] of modular forms, and relate this deformation to the Weyl-Moyal product. We also show that the projective structure introduced by Connes and Moscovici is equivalent to the existence of certain geometric data in the case of foliation groupoids. Using the methods developed by the second author [X. Tang, Deformation quantization of pseudo (symplectic) Poisson groupoids, Geom. Funct. Anal. 16 (3) (2006) 731-766], we reconstruct a universal deformation formula of the Hopf algebra H₁ associated to codimension one foliations. In the end, we prove that the first Rankin-Cohen bracket RC₁ defines a noncommutative Poisson structure for an arbitrary H₁ action.     

Superspecial abelian varieties and the Eichler Basis Problem for Hilbert modular forms

Let p be an unramified prime in a totally real field L such that h⁺(L)=1. Our main result shows that Hilbert modular newforms of parallel weight two for Γ₀(p) can be constructed naturally, via classical theta series, from modules of isogenies of superspecial abelian varieties with real multiplication on a Hilbert moduli space. This may be viewed as a geometric reinterpretation of the Eichler Basis Problem for Hilbert modular forms.     

On mod <Emphasis Type="Italic">p</Emphasis> properties of Siegel modular forms

We prove two results on mod p properties of Siegel modular forms. First, we use theta series in order to construct of a Siegel modular form of weight p−1 which is congruent to 1 mod p. Second, we define a theta operator on q-expansions and show that the algebra of Siegel modular forms mod p is stable under , by exploiting the relation between and generalized Rankin-Cohen brackets.     

Differential operators on Hermitian Jacobi forms

We study multilinear differential operators on a space of Hermitian Jacobi forms as well as on a space of Hermitian modular forms of degree 2. First we define a heat operator and construct multilinear differential operators on a space of Hermitian Jacobi forms of degree 2. As a special case of these operators, we also study Rankin-Cohen type differential operators on a space of Hermitian Jacobi forms. And we construct multilinear differential operators on a space of Hermitian modular forms of degree 2 as an application of multilinear differential operators on Hermitian Jacobi forms.     

Dirichlet series of Rankin-Cohen brackets

Given modular forms f and g of weights k and ?, respectively, their Rankin-Cohen bracket corresponding to a nonnegative integer n is a modular form of weight k+?+2n, and it is given as a linear combination of the products of the form f(r)g(n−r) for 0?r?n. We use a correspondence between quasimodular forms and sequences of modular forms to express the Dirichlet series of a product of derivatives of modular forms as a linear combination of the Dirichlet series of Rankin-Cohen brackets.     

Restricting Hecke-Siegel operators to Jacobi modular forms

We compute the action of Hecke operators on Jacobi forms of “Siegel degree” n and m×m index M, provided 1?j?n−m. We find they are restrictions of Hecke operators on Siegel modular forms, and we compute their action on Fourier coefficients. Then we restrict the Hecke-Siegel operators T(p), Tj(p²) (n−m<j?n) to Jacobi forms of Siegel degree n, compute their action on Fourier coefficients and on indices, and produce lifts from Jacobi forms of index M to Jacobi forms of index M^′ where detM|detM^′. Finally, we present an explicit choice of matrices for the action of the Hecke operators on Siegel modular forms, and for their restrictions to Jacobi modular forms.     

Rankin-Cohen Brackets and Invariant Theory

Using maps due to Ozeki and Broué-Enguehard between graded spaces of invariants for certain finite groups and the algebra of modular forms of even weight we equip these invariants spaces with a differential operator which gives them the structure of a Rankin-Cohen algebra. A direct interpretation of the Rankin-Cohen bracket in terms of transvectant for the group SL(2, C) is given.     

Differential eigenforms

The aim of this paper is to show how differential characters of Abelian varieties (in the sense of [A. Buium, Differential characters of Abelian varieties over p-adic fields, Invent. Math. 122 (1995) 309-340]) can be used to construct differential modular forms of weight 0 and order 2 (in the sense of [A. Buium, Differential modular forms, Crelle J. 520 (2000) 95-167]) which are eigenvectors of Hecke operators. These differential modular forms will have “essentially the same” eigenvalues as certain classical complex eigenforms of weight 2 (and order 0).     

On the Fourier coefficients of Hilbert-Maass wave forms of half integral weight over arbitrary algebraic number fields

The purpose of this paper is to derive a generalization of Shimura's results concerning Fourier coefficients of Hilbert modular forms of half integral weight over total real number fields in the case of Hilbert-Maass wave forms over algebraic number fields by following the Shimura's method. Employing theta functions, we shall construct the Shimura correspondence Ψ_τ from Hilbert-Maass wave forms f of half integral weight over algebraic number fields to Hilbert-Maass wave forms of integral weight over algebraic number fields. We shall determine explicitly the Fourier coefficients of in terms of these f. Moreover, under some assumptions about f concerning the multiplicity one theorem with respect to Hecke operators, we shall establish an explicit connection between the square of Fourier coefficients of f and the central value of quadratic twisted L-series associated with the image of f.     

Symbolic Computation for Rankin-Cohen Differential Algebras: A Case Study

Don Zagier defined a “Rankin-Cohen algebra”, motivated by the study of differential operators that send modular forms to modular forms. We devised an algorithm that computes the result of the differentiation given by the modular forms that correspond to higher-order Wronskians over Klein’s quartic curve, which are modular forms of arbitrarily high degree canonically attached to the curve; this tool is potentially useful for finding commutative rings of differential operators.     

Some formulas for the coefficients of Drinfeld modular forms

We obtain some formulas for t-expansion coefficients of meromorphic Drinfeld modular forms for GL₂(F_q[T]). Let j(z) be the Drinfeld modular invariant. As an application we show that the values of j(z) at points in the divisor of Drinfeld modular forms for GL₂(F_q[T]) are algebraic over F_q(T).     

Hecke operators on Drinfeld cusp forms

In this paper, we study the Drinfeld cusp forms for Γ₁(T) and Γ(T) using Teitelbaum's interpretation as harmonic cocycles. We obtain explicit eigenvalues of Hecke operators associated to degree one prime ideals acting on the cusp forms for Γ₁(T) of small weights and conclude that these Hecke operators are simultaneously diagonalizable. We also show that the Hecke operators are not diagonalizable in general for Γ₁(T) of large weights, and not for Γ(T) even of small weights. The Hecke eigenvalues on cusp forms for Γ(T) with small weights are determined and the eigenspaces characterized.     

Vector-valued Jacobi-like forms

We introduce vector-valued Jacobi-like forms associated to a representation of a discrete subgroup in 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.     

On Atkin and Swinnerton-Dyer congruence relations (3)

It is now well known that Hecke operators defined classically act trivially on genuine cuspforms for noncongruence subgroups of SL₂(Z). Atkin and Swinnerton-Dyer speculated the existence of p-adic Hecke operators so that the Fourier coefficients of their eigenfunctions satisfy three-term congruence recursions. In the previous two papers with the same title ([W.C. Li, L. Long, Z. Yang, On Atkin and Swinnerton-Dyer congruence relations, J. Number Theory 113 (1) (2005) 117-148] by W.C. Li, L. Long, Z. Yang and [A.O.L. Atkin, W.C. Li, L. Long, On Atkin and Swinnerton-Dyer congruence relations (2), Math. Ann. 340 (2) (2008) 335-358] by A.O.L. Atkin, W.C. Li, L. Long), the authors have studied two exceptional spaces of noncongruence cuspforms where almost all p-adic Hecke operators can be diagonalized simultaneously or semi-simultaneously. Moreover, it is shown that the l-adic Scholl representations attached to these spaces are modular in the sense that they are isomorphic, up to semisimplification, to the l-adic representations arising from classical automorphic forms.In this paper, we study an infinite family of spaces of noncongruence cuspforms (which includes the cases in [W.C. Li, L. Long, Z. Yang, On Atkin and Swinnerton-Dyer congruence relations, J. Number Theory 113 (1) (2005) 117-148; A.O.L. Atkin, W.C. Li, L. Long, On Atkin and Swinnerton-Dyer congruence relations (2), Math. Ann. 340 (2) (2008) 335-358]) under a general setting. It is shown that for each space in this family there exists a fixed basis so that the Fourier coefficients of each basis element satisfy certain weaker three-term congruence recursions. For a new case in this family, we will exhibit that the attached l-adic Scholl representations are modular and the p-adic Hecke operators can be diagonalized semi-simultaneously.     

On Rankin-Cohen brackets for Siegel modular forms

We determine an explicit formula for a Rankin-Cohen bracket for Siegel modular forms of degree on a certain subgroup of the symplectic group. Moreover, we lift that bracket via a Poincaré series to a Siegel cusp form on the full symplectic group.

     

Slopes of integral lattices

We use the dyadic trace to define the concept of slope for integral lattices. We present an introduction to the theory of the slope invariant. The main theorem states that a Siegel modular cusp form f of slope strictly less than the slope of an integral lattice with Gram matrix s satisfies f(sτ)=0 for all τ in the upper half plane. We compute the dyadic trace and the slope of each root lattice and we give applications to Siegel modular cusp forms.