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On the rank of abelian varieties over ample fields

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2010

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Petersen, Sebastian

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http://nbn-resolving.de/urn:nbn:de:bsz:352-127463
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https://doi.org/10.1142/S1793042110003071
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International Journal of Number Theory. 2010, 06(03), pp. 579-586. ISSN 1793-0421. Available under: doi: 10.1142/S1793042110003071

Zusammenfassung

A field K is called ample if every smooth K-curve that has a K-rational point has infinitely many of them. We prove two theorems to support the following conjecture, which is inspired by classical infinite rank results: Every non-zero Abelian variety A over an ample field K which is not algebraic over a finite field has infinite rank. First, the ℤ(p)-module A(K) ⊗ ℤ(p) is not finitely generated, where p is the characteristic of K. In particular, the conjecture holds for fields of characteristic zero. Second, if K is an infinite finitely generated field and S is a finite set of local primes of K, then every Abelian variety over K acquires infinite rank over certain subfields of the maximal totally S-adic Galois extension of K. This strengthens a recent infinite rank result of Geyer and Jarden.

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Fachgebiet (DDC)
510 Mathematik

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Abelian variety, infinite rank, ample field

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ISO 690FEHM, Arno, Sebastian PETERSEN, 2010. On the rank of abelian varieties over ample fields. In: International Journal of Number Theory. 2010, 06(03), pp. 579-586. ISSN 1793-0421. Available under: doi: 10.1142/S1793042110003071
BibTex
@article{Fehm2010abeli-12746,
  year={2010},
  doi={10.1142/S1793042110003071},
  title={On the rank of abelian varieties over ample fields},
  number={03},
  volume={06},
  issn={1793-0421},
  journal={International Journal of Number Theory},
  pages={579--586},
  author={Fehm, Arno and Petersen, Sebastian}
}
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