arXiv subjects

Younes Nikdelan

Publications and source records attributed to Younes Nikdelan.

9 recordsLinked to original sources

arXiv2025

About quasi-modular forms, differential operators and Rankin--Cohen algebras

We establish sufficient conditions, involving Rankin--Cohen (RC) brackets, under which certain combinations of meromorphic quasi-modular forms and their derivatives yield meromorphic modular forms. To achieve this, we adopt an algebraic perspective by working within the framework of RC algebras. First, we prove that any canonical RC algebra, whose underlying graded algebra is of type $\frac{1}{N}\mathbb{Z}$ with $N\in \mathbb{N}$, is a sub-RC algebra of a standard RC algebra. We then present and prove an algebraic formulation of results stating that specific combinations of the quasi-modular form $E_2$ with either other modular forms or itself, along with their derivatives, result in modular forms. Next, we provide equivalent formulations of these results in terms of the Lie algebra $\mathfrak{sl}_2(\mathbb{C})$ and Ramanujan systems of RC type. Finally, we discuss applications not only to meromorphic quasi-modular forms but also to Calabi--Yau quasi-modular forms.

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arXiv2023

Ramanujan systems of Rankin-Cohen type and hyperbolic triangles

In the first part of the paper we characterize certain systems of first order nonlinear differential equations whose space of solutions is an $\mathfrak{sl}_2(\mathbb{C})$-module. We prove that such systems, called Ramanujan systems of Rankin-Cohen type, have a special shape and are precisely the ones whose solution space admits a Rankin-Cohen structure. In the second part of the paper we consider triangle groups $Δ(n,m,\infty)$. By means of modular embeddings, we associate to every such group a number of systems of non linear ODEs whose solutions are algebraically independent twisted modular forms. In particular, all rational weight modular forms on $Δ(n,m,\infty)$ are generated by the solutions of one such system (which is of Rankin-Cohen type). As a corollary we find new relations for the Gauss hypergeometric function evaluated at functions on the upper half-plane. To demonstrate the power of our approach in the non classical setting, we construct the space of integral weight twisted modular form on $Δ(2,5,\infty)$ from solutions of systems of nonlinear ODEs.

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arXiv2022

Rankin-Cohen brackets for Calabi-Yau modular forms

For any positive integer $n$, we introduce a quasi-homogeneous vector field $\textsf{D}$ of degree $2$ on a moduli space $\textsf{T}$ of enhanced Calabi-Yau $n$-folds arising from the Dwork family. By Calabi-Yau quasi-modular forms for Dwork family we mean the elements of the graded $\mathbb{C}$-algebra $\widetilde{\mathcal{M}}$ generated by the components of a particular solution of $\textsf{D}$, which are provided with natural weight. Using $\textsf{D}$ we introduce the derivation $\mathcal{D}$ and the Ramanujan-Serre type derivation $\partial$ on $\widetilde{\mathcal{M}}$. We show that they are degree $2$ differential operators and there exists a proper subspace $\mathcal{M}\subset \widetilde{\mathcal{M}}$, called the space of Calabi-Yau modular forms, which is closed under $\partial$. Using the derivation $\mathcal{D}$, we define the Rankin-Cohen brackets for Calabi-Yau quasi-modular forms and prove that the subspace generated by the positive weight elements of $\mathcal{M}$ is closed under the Rankin-Cohen brackets.

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arXiv2021

Gauss-Manin Connection in Disguise: Dwork Family

We study the moduli space $\textsf{T}$ of the Calabi-Yau $n$-folds arising from the Dwork family and enhanced with bases of the $n$-th de Rham cohomology with constant cup product and compatible with Hodge filtration. We also describe a unique vector field $\textsf{R}$ in $\textsf{T}$ which contracted with the Gauss-Manin connection gives an upper triangular matrix with some non-constant entries which are natural generalizations of Yukawa couplings. For $n=1,2$ we compute explicit expressions of $\textsf{R}$ and give a solution of $\textsf{R}$ in terms of quasi-modular forms. The moduli space $\textsf{T}$ is an affine variety and for $n=4$ we give explicit coordinate system for $\textsf{T}$ and compute the vector field $\textsf{R}$ and the $q$-expnasion of its solution.

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arXiv2021

Ramanujan-type systems of nonlinear ODEs for $Γ_0(2)$ and $Γ_0(3)$

This paper aims to introduce two systems of nonlinear ordinary differential equations whose solution components generate the graded algebra of quasi-modular forms on Hecke congruence subgroups $Γ_0(2)$ and $Γ_0(3)$. Using these systems, we provide the generated graded algebras with an $\mathfrak{sl}_2(\mathbb{C})$-module structure. As applications, we introduce Ramanujan-type tau functions for $Γ_0(2)$ and $Γ_0(3)$, and obtain some interesting and non-trivial recurrence and congruence relations.

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arXiv2020

Modular vector fields attached to Dwork family: $\mathfrak{sl}_2(\mathbb{C})$ Lie algebra

This paper aims to show that a certain moduli space $\textsf{T}$, which arises from the so-called Dwork family of Calabi-Yau $n$-folds, carries a special complex Lie algebra containing a copy of $\mathfrak{sl}_2(\mathbb{C})$. In order to achieve this goal, we introduce an algebraic group $\sf G$ acting from the right on $\textsf{T}$ and describe its Lie algebra ${\rm Lie}({\sf G})$. We observe that ${\rm Lie}({\sf G})$ is isomorphic to a Lie subalgebra of the space of the vector fields on $\textsf{T}$. In this way, it turns out that ${\rm Lie}({\sf G})$ and the modular vector field ${\sf R}$ generate another Lie algebra $\mathfrak{G}$, called AMSY-Lie algebra, satisfying $\dim (\mathfrak{G})=\dim (\textsf{T})$. We find a copy of $\mathfrak{sl}_2(\mathbb{C})$ containing ${\sf R}$ as a Lie subalgebra of $\mathfrak{G}$. The proofs are based on an algebraic method calling "Gauss-Manin connection in disguise". Some explicit examples for $n=1,2,3,4$ are stated as well.

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arXiv2018

Product formulas for weight two newforms

For a weight two newform $f$ attached to an elliptic curve $E$ defined over rational numbers we write $f=q\prod_{n=1}^\infty (1-q^n)^{g_n}, \ g_n\in\Z$ and we observe that for some special elliptic curves $g_n$ is an increasing sequence of positive integers.

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arXiv2018

Manifold Ways to Darboux-Halphen System

Many distinct problems give birth to Darboux-Halphen system of differential equations and here we review some of them. The first is the classical problem presented by Darboux and later solved by Halphen concerning finding infinite number of double orthogonal surfaces in $\mathbb{R}^3$. The second is a problem in general relativity about gravitational instanton in Bianchi IX metric space. The third problem stems from the new take on the moduli of enhanced elliptic curves called Gauss-Manin connection in disguise developed by one of the authors and finally in the last problem Darboux-Halphen system emerges from the associative algebra on the tangent space of a Frobenius manifold.

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arXiv2014

Darboux-Halphen-Ramanujan Vector Field on a Moduli of Calabi-Yau Manifolds

In this paper we obtain an ordinary differential equation ${\sf H}$ from a Picard-Fuchs equation associated with a nowhere vanishing holomorphic $n$-form. We work on a moduli space ${\sf T }$ constructed from a Calabi-Yau $n$-fold $W$ together with a basis of the middle complex de Rham cohomology of $W$. We verify the existence of a unique vector field ${\sf H}$ on ${\sf T }$ such that its composition with the Gauss-Manin connection satisfies certain properties. The ordinary differential equation given by ${\sf H}$ is a generalization of differential equations introduced by Darboux, Halphen and Ramanujan.

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