Anharmonic solutions to the Riccati equation and elliptic modular functions. (English) Zbl 1435.12004

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 \(\mathbb{C}(\mathcal{P}, \mathcal{P}')\).


12H20 Abstract differential equations
11F03 Modular and automorphic functions
33E05 Elliptic functions and integrals
12H05 Differential algebra
34M15 Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical) of ordinary differential equations in the complex domain
34A26 Geometric methods in ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
34C14 Symmetries, invariants of ordinary differential equations
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