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Let $E/\mathbb{Q}$ be an elliptic curve with finitely many rational points.

What are some examples where at least one rational point has large coordinates (compared to the height of $E$)?

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    $\begingroup$ Can't happen if you put $E$ in Weierstrass form $y^2 = P(x)$ because the 2-torsion points are just $(x,0)$ where $x$ is one of the zeros of $P$. $\endgroup$ May 16, 2021 at 16:43
  • $\begingroup$ @NoamD.Elkies I see. What if we just require the rank to be zero? $\endgroup$
    – folenn
    May 16, 2021 at 17:01
  • $\begingroup$ Rank 0 means the points are torsion, and there are only finitely many by Mordell's theorem. If $E$ is in Weierstrass form with integer coefficients and $(x:y:1)$ is a torsion point, then either $y=0$ or it divides the discriminant (Lutz-Nagell). $\endgroup$
    – user166831
    May 16, 2021 at 18:27

1 Answer 1

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Generalizing Noam's comment and expanding a bit on anon's comment, if you put your curve in Weierstrass form $$ y^2 = P(x) = x^3+ax^2+bx+c\quad\text{with $a,b,c\in\mathbb Z$,} $$ then the Lutz-Nagell theorem says two things:

  • Any torsion point $(x,y)\in E(\mathbb Q)$ satisfies $x,y\in\mathbb Z$.
  • For such a point, either $y=0$, which case the point has order 2, or else $y^2$ divides the discriminant of $P(x)$.

In particular, if you put $E$ in short Weierstrass form $$y^2=x^3+Ax+B\quad\text{ with $A,B\in\mathbb Z$},$$ then any torsion point $(x,y)\in E(\mathbb Q)$ of order at least $3$ has integer coordinates satisfying $y^2\mid16(4A^3+27B^2)$. So if the point has large coordinates, then so does $E$.

An alternative version of this is to note that torsion points have canonical height $0$, and the canonical height differs from the Weil height by a quantity bounded in terms of the coefficients. Explicitly, there are absolute constants $c_1>0$ and $c_2$ so that for all elliptic curves as above and all rational points $(x,y)$, we have $$ \Bigl| h([x,y,1]) - \hat h([x,y,1]) \Bigr| \le c_1 h([A^3,B^2,1]) + c_2.$$ In particular, if $(x,y)$ is a torsion point, then $$ h([A^3,B^2,1]) \ge c_1^{-1} h([x,y,1]) - c_1^{-1}c_2. $$ One can find various explicit values for $c_1$ and $c_2$ in the literature.

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