The average number of integral points on the congruent number curves

Abstract

We show that the total number of non-torsion integral points on the curves E D : y 2 = x 3 − D 2 x, where D ranges over positive squarefree integers less than N , is ≪ N (log N ) −1/8 .

Integral points on cubic twists of Mordell curves

“…Then since 1 ≤ |c| ≤ N 3 4 , there are ≪ log N log ǫ k choices of the exponent of ǫ k . We conclude that for any fixed |c| ≤ N 2 3 , the number of possible pairs (S, T ) is bounded by ≪ k (log N) 2 . Therefore the number of (b, c) with |c| ≤ N…”

“…We start by showing that we can transform f P to integral cubic forms with smaller discriminants. A similar discriminant-reducing lemma for binary quartic forms can be found in [3]. Lemma 4.1.…”

“…3 by [14, Theorem 1]. Therefore each GL 2 (Z)-equivalence class of F is associated to ≪ (N/g) 2 3 integral points. Observe that a, h, u, g 0 , g 1 together determines the GL 2 (Z)-equivalence class of F , and g 0 is determined by a, h, u, g 1 by (5.2).…”

“…We will give some heuristics in Section 2, which suggest that the orders of magnitude of the quantities in Theorem 1.1 and Theorem 1.2 should be ≍ k N 2 3 , though when k is a square the points (0, ± √ kB) should be removed. Counting integer solutions to Mordell equation is related to counting the number of elliptic curves over Q with bounded discriminants.…”