The second moment of the number of integral points on elliptic curves is bounded

Abstract

In this paper, we show that the second moment of the number of integral points on elliptic curves over Q is bounded. In particular, we prove that, for any 0 < s < log 2 5 = 2.3219 . . ., the s-th moment of the number of integral points is bounded for many families of elliptic curvese.g., for the family of all integral short Weierstrass curves ordered by naive height, for the family of only minimal such Weierstrass curves, for the family of semistable curves, or for subfamilies thereof defined by finitely many congruence conditions. For certain other families of elliptic curves, such as those with a marked point or a marked 2-torsion point, the same methods show that for 0 < s < log 2 3 = 1.5850 . . ., the s-th moment of the number of integral points is bounded.The main new ingredient in our proof is an upper bound on the number of integral points on an affine integral Weierstrass model of an elliptic curve depending only on the rank of the curve and the number of square divisors of the discriminant. We obtain the bound by studying a bijection first observed by Mordell between integral points on these curves and certain types of binary quartic forms. The theorems on moments then follow from Hölder's inequality, analytic techniques, and results on bounds on the average sizes of Selmer groups in the families.Remark 1.2. If v p (∆ A,B ) = 2 or 3, the factor in Theorem 1.1 for p may be improved to 4 (rather than 5). This results from a slightly more careful analysis of the p-adic argument at the end of Bombieri-Schmidt [BS87]; see Remark 3.3.

The average number of integral points on the congruent number curves

“…The following correspondence is given by Mordell [16,Chapter 25] (or see [3,Section 2.3] for a modern interpretation).…”

“…where D denotes a positive squarefree integer. We are interested in the set of integral points E D (Z) := (x, y) ∈ Z 2 : y 2 = x 3 + AD 2 x + BD 3 .…”

“…Let E be an elliptic curve given by the model y 2 = f (x) where f ∈ Z[x] and deg f = 3. Then #{D ∈ D N : Dy 2 = f (x) has a non-trivial integral point} ∼ C f N 1 3 , where C f is some explicit constant determined by f . In this direction, Matschke and Mudigonda [15] handled the case when f (x) is reducible, assuming the abc conjecture.…”

Note on a theorem of Professor X

“…2 In this note we rectify the situation. 1 By this common and incorrect abbreviation we really mean integral solutions of y 2 = f (x). Siegel's 1926 proof does not control integral solutions of e.g.…”

“…In the elliptic curve case an even stronger bound is available because one can execute the whole descent over K (following Mordell) [1] 3 . In the superelliptic (y m = f (x), m > 2) case as usual one does not need to execute a 3-descent on G m so the bound also improves.…”