Integral points on the congruent number curve

Chan, Stephanie

doi:10.1090/tran/8732

Integral points on the congruent number curve
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Trans. Amer. Math. Soc. 375 (2022), 6675-6700 Request permission

Abstract:

We study integral points on the quadratic twists $\mathcal {E}_D:y^2=x^3-D^2x$ of the congruent number curve. We give upper bounds on the number of integral points in each coset of $2\mathcal {E}_D(\mathbb {Q})$ in $\mathcal {E}_D(\mathbb {Q})$ and show that their total is $\ll (3.8)^{\operatorname {rank} \mathcal {E}_D(\mathbb {Q})}$. 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 $aX^2-bY^2=d,\ bY^2-cZ^2=d$ for pairwise coprime positive integers $a,b,c,d$, has at most $\ll (3.6)^{\omega (abcd)}$ integer solutions.

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