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Journal of the Ramanujan Mathematical Society

Volume 34, Issue 2, June 2019  pp. 263–269.

Polynomial Pell equations P(x)2-(x2m+ax+b) Q(x)2 = 1 and associated hyperelliptic curves

Authors:  Tomasz Jedrzejak
Author institution:University of Szczecin, Faculty of Mathematics and Physics, Institute of Mathematics, Wielkopolska 15, 70-451 Szczecin, Poland

Summary:  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 (Cm,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.

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