Polynomial Pell equations P x2 – x2m + b Q x2 and associated hyperelliptic curves

Abstract

The title equations are connected with Jacobians of hyperelliptic curves Cm,a,b : y2 = x 2m + ax + b defined over (Ql. More precisely, these equations have a nontrivial solution if and only if the class of the divisor oo+ - oo- is a torsion point in Jacobian Jac(Cm,a,b), where oo+ and oo- are two points at infinity in Cm,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, form = 2, 3 for almost all (a, b) with ab i= 0, these equations have no nontrivial solutions. We also give infinitely many explicit examples when nontrivial solution does not exist.