The Ring of Quasimodular Forms for a Cocompact Group

Modular Linear Differential Operators and Generalized Rankin-Cohen Brackets

The aim in this paper is to give expressions for modular linear differential operators 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 …

Quasimodular forms, Jacobi-like forms, and pseudodifferential operators

We study various properties of quasimodular forms by using their connections with Jacobi-like forms and pseudodifferential operators. Such connections are made by identifying quasimodular forms for a discrete subgroup $\G$ of $SL(2, \bR)$ with certain polynomials over the ring of holomorphic functions of the Poincaré upper half plane that are $\G$-invariant. We consider a surjective map from Jacobi-like forms to quasimodular forms and prove that it has a right inverse, which may be regarded as a lifting from quasimodular forms to Jacobi-like forms. We use such liftings to study Lie brackets an…

利用拟模形式计算双曲函数的倒数和

Evaluation of reciprocal sums of hyperbolic functions using quasimodular forms

本文研究了八族双曲函数无穷级数，提出利用复乘理论、模形式与拟模形式环及其微分算子计算其在 CM 点值的系统方法，并据此导出了 Fibonacci 数广义倒数和的代数无关性结果。

The Eichler integral of $E_2$ and $q$-brackets of $t$-hook functions

For functions $f: \mathcal{P}\rightarrow \mathbb{C}$ on partitions, Bloch and Okounkov defined a power series $\langle f\rangle_q$ that is the "weighted average" of $f$. As Fourier series in $q=e^{2πi z}$, such $q$-brackets generate the ring of quasimodular forms, and the modular forms that are powers of Dedekind's eta-function. Using work of Berndt and Han, we build modular objects from $$ f_t(λ):= t\sum_{h\in \mathcal{H}_t(λ)}\frac{1}{h^2}, $$ weighted sums over partition hook numbers that are multiples of $t$. We find that $\langle f_t \rangle_q$ is the Eichler integral of $(1-E_2(tz))/24,$…

Estimates of Hilbert modular cusp forms of half-integral and integral weight

Let $Γ$ be a cocompact, discrete, and irreducible subgroup of $\mathrm{PSL}_{2}(\mathbb{R})^{n}$. Let $ν$ be a unitary character of $Γ$. For $k\in1\slash 2\,\mathbb{Z}$, let $\sknu$ denote the complex vector space of cusp forms of weight-$\tk=\k$ and nebentypus $ν^{2k}$ with respect to $Γ$. We assume that $ω_{X,ν}$, the line bundle of cusp forms of weight-$\tilde{1\slash 2}:=(1\slash 2,\ldots,1\slash2)$ with nebentypus $ν$ over $X$ exists. Let $\lbrace f_{1},\ldots,f_{j_{\tk}} \rbrace$ denote an orthonormal basis of $\sknu$. In this article, we show that as $k\rightarrow \infty$, the sum $\sum…