Polynomial Pell Equations P(x)2 − (x2m + Ax + B)q(x)2 = 1 and Associated Hyperelliptic Curves

The title equations are connected with Jacobians of hyperelliptic curves C_m,a,b : y² = x^2m + ax + b defined over Q. More precisely, these equations have a nontrivial solution if and only if the class of the divisor ∞⁺ − ∞⁻ is a torsion point in Jacobian Jac(C_m,a,b), where ∞⁺ and ∞⁻ are two points at infinity in C_m,a,b. We show that if ab = 0 then the title equations have nontrivial solutions (and we write explicit formulae). On the other hand, we prove that for any m > 1 there exist infinitely many pairs (a, b) such that our equations have no nontrivial solutions. Moreover, for m = 2, 3 for almost all (a, b) with ab ≠ 0, these equations have no nontrivial solutions. We also give infinitely many explicit examples when nontrivial solution does not exist.