On the torsion values for sections of an elliptic scheme

TL;DR

This paper investigates the distribution of torsion points on sections of elliptic schemes over curves, relating it to heights, measures, and multiplicities, and establishes finiteness and effective results akin to Siegel's theorem.

Contribution

It introduces a new integral formula connecting torsion point distribution with the Betti measure and proves finiteness results for high-multiplicity torsion points, linking to Diophantine approximation.

Findings

Distribution of torsion points relates to the canonical height.

Finiteness theorems for points with higher-than-expected multiplicity.

Effective bounds analogous to Siegel's theorem on integral points.

Abstract

We shall consider sections of an elliptic scheme $\mathcal{E}$ over a(n affine) base curve $B$, and study the points of $B$ where the section takes a torsion value. In particular, we shall relate the distribution in $B$ of these points with the canonical height of the section, proving an integral formula involving a measure on $B$ coming from the so-called Betti map of the section. We shall show that this measure is the same appearing in dynamical issues related to the section. This analysis will also involve the multiplicity with which a torsion value is attained, which is an independent problem. We shall prove finiteness theorems for the points where the multiplicity is higher than expected. Such multiplicity has also a relation with Diophantine Approximation and quasi-integral points on $\mathcal{E}$ (over the affine ring of $B$), and in the last part of the paper we shall exploit this viewpoint, proving an effective result in the spirit of Siegel's theorem on integral points.