Rankin Cohen Brackets Research Articles

Hilbert Poincaré series and kernels for products of L-functions

We study Hilbert Poincaré series associated to general seed functions and construct Cohen’s kernels and double Eisenstein series as series of Hilbert Poincaré series. Then we calculate the Rankin–Cohen brackets of Hilbert Poincaré series and Hilbert modular forms and extend Zagier’s kernel formula to totally real number fields. Finally, we show that the Rankin–Cohen brackets of two different types of Eisenstein series are special values of double Eisenstein series up to a constant.

Simple vertex algebras arising from congruence subgroups

For any congruence subgroup Γ, we consider the vertex algebra of Γ-invariant global sections of chiral de Rham complex on the upper half plane that are meromorphic at the cusps. We give a description of the linear structure of the Γ-invariant vertex algebra by exhibiting a linear basis determined by meromorphic modular forms, and generalize the Rankin-Cohen bracket of modular forms to meromorphic modular forms. We also show that the Γ-invariant vertex algebra is simple.

On Rankin–Cohen brackets of Jacobi forms and Jacobi Poincaré series of matrix index

On Rankin–Cohen brackets of Jacobi forms and Jacobi Poincaré series of matrix index

A Selberg identity for the Shimura lift

AbstractWe first prove a Selberg‐type identity for the Shimura lift of the Rankin–Cohen bracket of a normalized Hecke eigenform and the theta function. We then discuss its relationship with the nonvanishing of central values of ‐functions associated to Hecke eigenforms.

Modular linear differential operators and generalized Rankin-Cohen brackets

The aim of this paper is to give expressions for modular linear differential operators (MLDOs) of any order. In particular, we show that they can all be described in terms of Rankin-Cohen brackets and a modified Rankin-Cohen bracket found by Kaneko and Koike. We also give more uniform descriptions of MLDOs in terms of canonically defined higher Serre derivatives and an extension of Rankin-Cohen brackets, as well as in terms of quasimodular forms and almost holomorphic modular forms. The last of these descriptions involves the holomorphic projection map. The paper also includes some general results on the theory of quasimodular forms on both cocompact and non-cocompact subgroups of S L 2 ( R ) SL_2(\mathbb {R}) , as well as a slight sharpening of a theorem of Martin and Royer on Rankin-Cohen brackets of quasimodular forms.

Branching problem for tensoring two Verma modules and its application to differential symmetry breaking operators

Kobayashi–Pevzner discovered in [Differential symmetry breaking operators: II. Rankin–Cohen operators for symmetric pairs, Selecta Math. (N.S.) 22(2) (2016) 847–911, MR 3477337] that the failure of the multiplicity-one property in the fusion rule of Verma modules of [Formula: see text] occurs exactly when the Rankin–Cohen brackets vanish, and classified all the corresponding parameters. In this paper, we provide yet another characterization for these parameters, and give a precise description of indecomposable components of the tensor product. Furthermore, we discuss when the tensor products of two Verma modules are isomorphic to each other for semisimple Lie algebras [Formula: see text].

Generating operators of symmetry breaking—From discrete to continuous

Based on the “generating operator” of the Rankin–Cohen brackets introduced in Kobayashi–Pevzner [arXiv:2306.16800], we present a method to construct various fundamental operators with continuous parameters such as invariant trilinear forms on infinite-dimensional representations, the Fourier and the Poisson transforms on the anti-de Sitter space, and integral symmetry breaking operators for the fusion rules, among others, out of a countable set of differential symmetry breaking operators.

Identities for Rankin–Cohen brackets, Racah coefficients and associativity

Identities for Rankin–Cohen brackets, Racah coefficients and associativity

Rankin–Cohen brackets of Jacobi forms and Jacobi Poincaré series

Rankin–Cohen brackets of Jacobi forms and Jacobi Poincaré series

Construction of modular differential equations by using Rankin-Cohen brackets

In this paper, we construct higher-order modular differential equations for elliptic modular forms, holomorphic Jacobi forms, and skew-holomorphic Jacobi forms by using the Rankin-Cohen brackets extended to the Eisenstein series of weight two. Rankin-Cohen brackets are typical tools in the construction of modular forms by differential operators. We introduce the extended Rankin-Cohen brackets for holomorphic or skew-holomorphic Jacobi forms paired with elliptic modular forms. They provide a unified method of construction of higher-order modular differential equations for the modular forms above. In particular, with these extended Rankin-Cohen brackets, we can reproduce the modular differential equations of previous studies.

Automorphic products that are singular modulo primes

We use Rankin–Cohen brackets on O(n,2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{O}(n, 2)$$\\end{document} to prove that the Fourier coefficients of reflective Borcherds products often satisfy congruences modulo certain primes.

Differentiating Siegel Modular Forms and the Moving Slope of 𝒜g

Abstract We study the cone of moving divisors on the moduli space ${\mathcal{A}}_{g}$ of principally polarized abelian varieties. Partly motivated by the generalized Rankin–Cohen bracket, we construct a non-linear holomorphic differential operator that sends Siegel modular forms to Siegel modular forms, and we apply it to produce new modular forms. Our construction recovers the known divisors of minimal moving slope on ${\mathcal{A}}_{g}$ for $g\leq 4$, and gives an explicit upper bound for the moving slope of ${\mathcal{A}}_{5}$ and a conjectural upper bound for the moving slope of ${\mathcal{A}}_{6}$.

Linear independence of even periods of modular forms

We show that if the dimension of the space of cuspforms is greater than or equal to three, then any three even periods are linearly independent. We also prove an asymptotic result for an arbitrary number of even periods. These results are achieved by studying the Rankin-Cohen brackets of Eisenstein series.

Rankin–Cohen brackets of vector valued Eisenstein series

Rankin–Cohen brackets of vector valued Eisenstein series

Modular forms via invariant theory

We discuss two simple but useful observations that allow the construction of modular forms from given ones using invariant theory. The first one deals with elliptic modular forms and their derivatives, and generalizes the Rankin-Cohen bracket, while the second one deals with vector-valued modular forms of genus greater than 1.

Explicit Sato–Tate type distribution for a family of K ⁢ 3 K3 surfaces

Abstract In the 1960s, Birch proved that the traces of Frobenius for elliptic curves taken at random over a large finite field is modeled by the semicircular distribution (i.e. the usual Sato–Tate for non-CM elliptic curves). In analogy with Birch’s result, a recent paper by Ono, the author, and Saikia proved that the limiting distribution of the normalized Frobenius traces A λ ⁢ ( p ) {A_{\lambda}(p)} of a certain family of K ⁢ 3 {K3} surfaces X λ {X_{\lambda}} with generic Picard rank 19 is the O ⁢ ( 3 ) {O(3)} distribution. This distribution, which we denote by 1 4 ⁢ π ⁢ f ⁢ ( t ) {\frac{1}{4\pi}f(t)} , is quite different from the semicircular distribution. It is supported on [ - 3 , 3 ] {[-3,3]} and has vertical asymptotes at t = ± 1 {t=\pm 1} . Here we make this result explicit. We prove that if p ≥ 5 {p\geq 5} is prime and - 3 ≤ a < b ≤ 3 {-3\leq a<b\leq 3} , then | # ⁢ { λ ∈ 𝔽 p : A λ ⁢ ( p ) ∈ [ a , b ] } p - 1 4 ⁢ π ⁢ ∫ a b f ⁢ ( t ) ⁢ 𝑑 t | ≤ 98.28 p 1 / 4 . \biggl{\lvert}\frac{\#\{\lambda\in\mathbb{F}_{p}:A_{\lambda}(p)\in[a,b]\}}{p}-% \frac{1}{4\pi}\int_{a}^{b}f(t)\,dt\biggr{\rvert}\leq\frac{98.28}{p^{1/4}}. As a consequence, we are able to determine when a finite field 𝔽 p {\mathbb{F}_{p}} is large enough for the discrete histograms to reach any given height near t = ± 1 {t=\pm 1} . To obtain these results, we make use of the theory of Rankin–Cohen brackets in the theory of harmonic Maass forms.

Rankin–Cohen brackets of Eisenstein series

We prove that Rankin–Cohen brackets of Eisenstein series generate the whole space of cuspforms in several cases. This generalizes the classical result that products of two Eisenstein series generate the whole space of modular forms.

Some Eichler-Selberg Trace Formulas

The Eichler-Selberg trace formulas express the traces of Hecke operators on a spaces of cusp forms in terms of weighted sums of Hurwitz-Kronecker class numbers. For cusp forms on $\text {\rm SL}_2(\mathbb{Z}),$ Zagier proved these formulas by cleverly making use of the weight 3/2 nonholomorphic Eisenstein series he discovered in the 1970s. The holomorphic part of this form, its so-called {\it mock modular form}, is the generating function for these class numbers. In this expository note we revisit Zagier's method, and we show how to obtain such formulas for congruence subgroups, working out the details for $\Gamma_0(2)$ and $\Gamma_0(4).$ The trace formulas fall out naturally from the computation of the Rankin-Cohen brackets of Zagier's mock modular form with specific theta functions.

Meromorphic quasi-modular forms and their L-functions

We investigate meromorphic quasi-modular forms and their L-functions. We study the space of meromorphic quasi-modular forms and Rankin–Cohen brackets of meromorphic modular forms. Then we define their L-functions by using multiple techniques of regularized integral. Explicit formulas for the L-functions are given.

Higher Siegel theta lifts on Lorentzian lattices, harmonic Maass forms, and Eichler–Selberg type relations

We investigate so-called “higher” Siegel theta lifts on Lorentzian lattices in the spirit of Bruinier–Ehlen–Yang and Bruinier–Schwagenscheidt. We give a series representation of the lift in terms of Gauss hypergeometric functions, and evaluate the lift as the constant term of a Fourier series involving the Rankin–Cohen bracket of harmonic Maass forms and theta functions. Using the higher Siegel lifts, we obtain a vector-valued analogue of Mertens’ result stating that the Rankin–Cohen bracket of the holomorphic part of a harmonic Maass form of weight $$\frac{3}{2}$$ and a unary theta function, plus a certain form, is a holomorphic modular form. As an application of these results, we offer a novel proof of a conjecture of Cohen which was originally proved by Mertens, as well as a novel proof of a theorem of Ahlgren and Kim, each in the scalar-valued case.