Chan, Stephanie;
(2022)
Integral points on the congruent number curve.
Transactions of the American Mathematical Society
, 375
(9)
pp. 6675-6700.
10.1090/tran/8732.
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Abstract
We study integral points on the quadratic twists ED : y 2 = x 3 − D2x of the congruent number curve. We give upper bounds on the number of integral points in each coset of 2ED(Q) in ED(Q) and show that their total is ≪ (3.8)rank ED(Q) . We further show that the average number of non-torsion integral points in this family is bounded above by 2. As an application we also deduce from our upper bounds that the system of simultaneous Pell equations aX2 − bY 2 = d, bY 2 − cZ2 = d for pairwise coprime positive integers a, b, c, d, has at most ≪ (3.6)ω(abcd) integer solutions.
| Type: | Article |
|---|---|
| Title: | Integral points on the congruent number curve |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1090/tran/8732 |
| Publisher version: | https://doi.org/10.1090/tran/8732 |
| Language: | English |
| Additional information: | This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions. |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10213469 |
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