6
$\begingroup$

Let $X\to \mathbb{P}^1$ be a non-isotrivial elliptic surface over $\mathbb{C}$ with a section and with $X$ a smooth projective connected surface over $\mathbb{C}$. Let $\sigma:\mathbb{P}^1\to X$ be a section of infinite order. Let $U\subset \mathbb{P}^1$ be the maximal dense open over which $f$ is smooth. The fibres over points in $U$ are naturally ellipic curves.

Does there exist a point $P$ in $U$ such that $\sigma(P)$ is torsion in the elliptic curve $X_P$?

Note that the non-isotriviality condition is necessary. Otherwise the answer is clearly negative.

Improve this question
$\endgroup$
2
  • 3
    $\begingroup$ Yes, there is are infinitely many such special parameters $P$ at which $\sigma(P)$ becomes torsion in its fibre. Moreover, all large enough orders are realizable for the torsion point. This comes by a version of the implicit function theorem. When the data is over the algebraic numbers, one further knows that these special parameters $P$ are a set of bounded height (so they are rather sparse). See Masser and Zannier's paper Torsion anomalous points and families of elliptic curves and, for a generalization, Prop. 3.1 in Habegger's paper Special points on fibered powers of elliptic surfaces. $\endgroup$
    – Vesselin Dimitrov
    Commented Apr 19, 2020 at 6:54
  • $\begingroup$ @VesselinDimitrov Thank you for this. Can you make this into an answer? $\endgroup$
    – Pat
    Commented May 3, 2020 at 8:16

0

Reset to default

You must log in to answer this question.