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

### MSC:

 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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