Transversality of sections on elliptic surfaces with applications to elliptic divisibility sequences and geography of surfaces

Ulmer, Douglas; Urzúa, Giancarlo

doi:10.1007/s00029-021-00747-x

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2023

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“…Very recently D. Ulmer and G. Urzua kindly informed us about their preprint[22] in which, among other things, they proved the finiteness instance in the case of a cyclic group, using somewhat different methods.…”

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confidence: 99%

On the torsion values for sections of an elliptic scheme

Corvaja¹,

Demeio²,

Masser³

et al. 2019

Preprint

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We shall consider sections of an elliptic scheme E over an 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 one which appears 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 E (over the affine ring of B), and in sections 5 and 6 of the paper we shall exploit this viewpoint, proving an effective result in the spirit of Siegel's theorem on integral points.

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confidence: 99%