Anharmonic solutions to the Riccati equation and elliptic modular functions
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 $\mathbb{C}(\mathcal{P}, \mathcal{P}')$.
scientific article; zbMATH DE number 6859565
scientific article