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https://gnanaganga.inflibnet.ac.in:8443/jspui/handle/123456789/6369
Title: | Polynomial Pell equations P x2 – x2m + b Q x2 and associated hyperelliptic curves |
Authors: | Tomasz Jedrzejak |
Issue Date: | 2019 |
Publisher: | Journal of the Ramanujan Mathematical Society |
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. |
URI: | http://gnanaganga.inflibnet.ac.in:8080/jspui/handle/123456789/6369 |
Appears in Collections: | Articles to be qced |
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