Elliptic surfaces and intersections of adelic -divisors

  • Laura DeMarco

    Harvard University, Cambridge, USA
  • Niki Myrto Mavraki

    Harvard University, Cambridge, USA
Elliptic surfaces and intersections of adelic $\mathbb{R}$-divisors cover
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Suppose is a non-isotrivial elliptic surface defined over a number field, for smooth projective curve . Let denote the function field and the associated elliptic curve over . In this article, we construct adelically metrized -divisors on the base curve over a number field, for each . We prove non-degeneracy of the Arakelov–Zhang intersection numbers , as a biquadratic form on . As a consequence, we have the following Bogomolov-type statement for the Néron–Tate height functions on the fibers of over : given points with , there exist an infinite sequence and small-height perturbations of specializations such that the set satisfies at least two independent linear relations for all , if and only if the points are linearly dependent in . This gives a new proof of results of Masser and Zannier (2010, 2012) and of Barroero and Capuano (2016) and extends our earlier 2020 results. In the Appendix, we prove an equidistribution theorem for adelically metrized -divisors on projective varieties (over a number field) using results of Moriwaki (2016), extending the equidistribution theorem of Yuan (2012).

Cite this article

Laura DeMarco, Niki Myrto Mavraki, Elliptic surfaces and intersections of adelic -divisors. J. Eur. Math. Soc. (2023), published online first

DOI 10.4171/JEMS/1354