Anharmonic solutions to the Riccati equation and elliptic modular functions
type: Item
instance of: scholarly article
title: Anharmonic solutions to the Riccati equation and elliptic modular functions
zbMATH Open document ID: 1435.12004
author: Ahmed Sebbar
author: Oumar Wone
published in: Forum Mathematicum
publication date: 2018-04-13T00:00:00Z
review text: The authors consider the differential field $(\mathbb{F}=\mathbb{C}(t),\delta =d/dt)$ and an irreducible algebraic equation (E) in one variable of degree $\geq 4$. They assume that a root of (E) is a solution to the Riccati differential equation $u'+B_0+B_1u+B_2u^2=0$, $B_j\in \mathbb{F}$. They construct a large class of polynomials as in (E), i.e. they prove the existence of a polynomial $F_n(x,y)\in \mathbb{C}(x)[y]$ such that for almost all $T\in \mathbb{F}\setminus \mathbb{C}$, all roots of the equation $F_n(x,T)=0$ are solutions to the same Riccati equation. They give an example of a degree 3 irreducible polynomial equation satisfied by certain weight 2 modular forms for the subgroup $\Gamma (2)$, whose roots satisfy a common Riccati equation on the differential field $(\mathbb{C}(E_2,E_4,E_6),d/d\tau )$, $E_j(\tau )$ being the Eisenstein series of weight $j$. These solutions are related to a Darboux-Halphen system. They also consider the question for which potentials $q\in \mathbb{C}(\mathcal{P}, \mathcal{P}')$ (where $\mathcal{P}$ is the classical Weierstrass function) does the Riccati equation $dY/dz+Y^2=q$ admit algebraic solutions over $\mathbb{C}(\mathcal{P}, \mathcal{P}')$.
Mathematics Subject Classification ID: 12H20
Mathematics Subject Classification ID: 11F03
Mathematics Subject Classification ID: 33E05
Mathematics Subject Classification ID: 12H05
Mathematics Subject Classification ID: 34M15
Mathematics Subject Classification ID: 34A26
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 34C14
zbMATH DE Number: 6859565
zbMATH Keywords: Riccati equation
zbMATH Keywords: Darboux-Halphen system
zbMATH Keywords: differential Galois theory
zbMATH Keywords: Darboux polynomial
zbMATH Keywords: modular forms
zbMATH Keywords: Cayley's process
zbMATH Keywords: Wedekind theorem
zbMATH Keywords: transvectants
reviewed by: Vladimir P. Kostov
MaRDI profile type: Publication
full work available at URL: forum-2017-0025
OpenAlex ID: W2744237929
cites work: On Second Order Linear Differential Equations with Algebraic Solutions on Algebraic Curves
cites work: On Second Order Linear Differential Equations with Algebraic Solutions
cites work: The 1-2-3 of modular forms. Lectures at a summer school in Nordfjordeid, Norway, June 2004
cites work: On automorphisms of Enriques surfaces
cites work: Q5768795
cites work: Transcendence of Jacobi's theta series
cites work: Q4112233
cites work: Q3235831
cites work: The invariant theory of binary forms
cites work: Q3709072
cites work: On algebraic differential equations satisfied by automorphic functions
cites work: Q4509277
cites work: Elementary First Integrals of Differential Equations
cites work: Q5164805
cites work: Q4790565
cites work: Q5804260
DOI: 10.1515/FORUM-2017-0025
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description: scientific article; zbMATH DE number 6859565