Journal of the Ramanujan Mathematical Society

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


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1 University of Szczecin, Institute of Mathematics, Wielkopolska 15, 70-451 Szczecin, Poland
     

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The title equations are connected with Jacobians of hyperelliptic curves Cm,a,b : y2 = x2m + 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 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, 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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  • Polynomial Pell Equations P(x)2 − (x2m + Ax + B)q(x)2 = 1 and Associated Hyperelliptic Curves

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Authors

Tomasz Jedrzejak
University of Szczecin, Institute of Mathematics, Wielkopolska 15, 70-451 Szczecin, Poland

Abstract


The title equations are connected with Jacobians of hyperelliptic curves Cm,a,b : y2 = x2m + 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 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, 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.

References