Anharmonic solutions to the Riccati equation and elliptic modular functions

Abstract

Abstract We study the irreducible algebraic equation x n + a 1 ⁢ x n - 1 + ⋯ + a n = 0 , with n ≥ 4 , x^{n}+a_{1}x^{n-1}+\cdots+a_{n}=0,\quad\text{with ${n\geq 4}$,} on the differential field ( 𝔽 = ℂ ⁢ ( t ) , δ = d d ⁢ t ) {(\mathbb{F}=\mathbb{C}(t),\delta=\frac{d}{dt})} . We assume that a root of the equation is a solution to the Riccati differential equation u ′ + B 0 + B 1 ⁢ u + B 2 ⁢ u 2 = 0 {u^{\prime}+B_{0}+B_{1}u+B_{2}u^{2}=0} , where B 0 {B_{0}} , B 1 {B_{1}} , B 2 {B_{2}} are in 𝔽 {\mathbb{F}} . We show how to construct a large class of polynomials as in the above algebraic equation, i.e., we prove that there exists a polynomial F n ⁢ ( x , y ) ∈ ℂ ⁢ ( x ) ⁢ [ y ] {F_{n}(x,y)\in\mathbb{C}(x)[y]} such that for almost T ∈ 𝔽 ∖ ℂ {T\in\mathbb{F}\setminus\mathbb{C}} , the algebraic equation F n ⁢ ( x , T ) = 0 {F_{n}(x,T)=0} is of the same type as the above stated algebraic equation. In other words, all its roots are solutions to the same Riccati equation. On the other hand, we give an example of a degree 3 irreducible polynomial equation satisfied by certain weight 2 modular forms for the subgroup Γ ⁢ ( 2 ) {\Gamma(2)} , whose roots satisfy a common Riccati equation on the differential field ( ℂ ⁢ ( E 2 , E 4 , E 6 ) , d d ⁢ τ ) {(\mathbb{C}(E_{2},E_{4},E_{6}),\frac{d}{d\tau})} , with E i ⁢ ( τ ) {E_{i}(\tau)} being the Eisenstein series of weight i. These solutions are related to a Darboux–Halphen system. Finally, we deal with the following problem: For which “potential” q ∈ ℂ ⁢ ( ℘ , ℘ ′ ) {q\in\mathbb{C}(\wp,\wp^{\prime})} does the Riccati equation d ⁢ Y d ⁢ z + Y 2 = q {\frac{dY}{dz}+Y^{2}=q} admit algebraic solutions over the differential field ℂ ⁢ ( ℘ , ℘ ′ ) {\mathbb{C}(\wp,\wp^{\prime})} , with ℘ {\wp} being the classical Weierstrass function? We study this problem via Darboux polynomials and invariant theory and show that the minimal polynomial Φ ⁢ ( x ) {\Phi(x)} of an algebraic solution u must have a vanishing fourth transvectant τ 4 ⁢ ( Φ ) ⁢ ( x ) {\tau_{4}(\Phi)(x)} .