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I am reading Silverman - Difference between Weil and canonical heights, and on page 739 in Example 7.1, the author is investigating the curve $E: y^2=x^3-x+1.$ The goal is to determine $E(\mathbb{Q})$. Here is an excerpt:

I do not understand the last part. What result is used to conclude that $x(P)\in\mathbb{Z}$ implies $x(R)\in\mathbb{Z}$? I know this isn't true in general, but does it hold for all curves with trivial torsion? Any help would be appreciated. Thanks.

Douglas Molin
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  • Because $(1,1)$ is a non-trivial non-singular point of $y^2=x^3-x+1\bmod q$ for all prime $q$ ? – reuns Oct 15 '20 at 10:18

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This is a general fact, not specific to this curve: if $E/\mathbb{Q}$ is an elliptic curve given by a Weierstrass model with coefficients in $\mathbb{Z}$, and $P\in E(\mathbb{Q})$ is a point of infinite order such that $x([m]P)\in\mathbb{Z}$, then $x(P)\in \mathbb{Z}$.

This result is Exercise 9.12 in Silverman's "The Arithmetic of Elliptic Curves." In the 2nd edition of the book there is a part (b) to this problem, which is technically a generalization, but it is essentially a hint, since it points out to Exercise 3.36 about elliptic divisibility sequences, which in turn point to division polynomials (to show that there is a divisibility relation among denominators of multiples of $P$).

Álvaro Lozano-Robledo
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