The relative Bogomolov conjecture for fibered products of elliptic curves

Lars Kühne

doi:10.1515/crelle-2022-0082

Article

The relative Bogomolov conjecture for fibered products of elliptic curves

Lars Kühne

Published/Copyright: January 27, 2023

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2023 Issue 795

Abstract

We deduce an analogue of the Bogomolov conjecture for non-degenerate subvarieties in fibered products of families of elliptic curves from the author’s recent theorem on equidistribution in families of abelian varieties. This generalizes results of DeMarco and Mavraki and improves certain results of Manin–Mumford type proven by Masser and Zannier to results of Bogomolov type, yielding the first results of this type for subvarieties of relative dimension > 1 in families of abelian varieties with trivial trace.

Funding statement: The author was supported by an Ambizione Grant of the Swiss National Science Foundation during the early stages of this project. He also received funding from the European Union Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 101027237.

Acknowledgements

The author thanks the anonymous referee for their attentive reading and their many suggestions that helped to improve the exposition substantially. He also thanks Laura DeMarco, Ziyang Gao, Philipp Habegger, Myrto Mavraki, and Fabien Pazuki for their advice, discussion and encouragement. Finally, he thanks Jakob Stix for pointing out some inaccuracies in the article.

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Received: 2021-03-10

Revised: 2022-10-18

Published Online: 2023-01-27

Published in Print: 2023-02-01

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https://doi.org/10.1515/crelle-2022-0082