Ramified covers of abelian varieties over torsion fields

Lior Bary-Soroker; Arno Fehm; Sebastian Petersen

doi:10.1515/crelle-2023-0077

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Ramified covers of abelian varieties over torsion fields

Lior Bary-Soroker , Arno Fehm und Sebastian Petersen

Veröffentlicht/Copyright: 1. November 2023

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Aus der Zeitschrift Journal für die reine und angewandte Mathematik (Crelles Journal) Band 2023 Heft 805

Abstract

We study rational points on ramified covers of abelian varieties over certain infinite Galois extensions of ℚ . In particular, we prove that every elliptic curve E over ℚ has the weak Hilbert property of Corvaja and Zannier both over the maximal abelian extension ℚ ab of ℚ , and over the field ℚ ⁢ ( A tor ) obtained by adjoining to ℚ all torsion points of some abelian variety A over ℚ .

Dedicated to Moshe Jarden on the occasion of his 80th birthday

Funding statement: Lior Bary-Soroker was supported by the Israel Science Foundation (grant no. 702/19). Sebastian Petersen was supported by a research grant UMO-2018/31/B/ST1/01474 of the National Centre of Sciences of Poland.

Acknowledgements

The authors would like to thank Daniele Garzoni for helpful discussions around [12], Cornelius Greither for interesting discussions around Lemma 2.11, Remy van Dobben de Bruyn for the suggestion to use the Hilbert scheme in the proof of Lemma 2.13, and the referee as well as Jakob Stix for helpful remarks on the submitted version. Part of this work was done while A.F. was a guest of Tel Aviv University, and he would like to thank the School of Mathematics for their hospitality.

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Received: 2022-09-24

Revised: 2023-08-17

Published Online: 2023-11-01

Published in Print: 2023-12-01

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https://doi.org/10.1515/crelle-2023-0077