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Diamonds in Torsion of Abelian Varieties

Part of: General field theory

Published online by Cambridge University Press:  10 February 2010

Moshe Jarden
Moshe Jarden
Affiliation:
School of Mathematics, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel (jarden@post.tau.ac.il
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Abstract

A theorem of Kuyk says that every Abelian extension of a Hilbertian field is Hilbertian. We conjecture that for an Abelian variety A defined over a Hilbertian field K every extension L of K in K(Ator) is Hilbertian. We prove our conjecture when K is a number field. The proof applies a result of Serre about l-torsion of Abelian varieties, information about l-adic analytic groups, and Haran's diamond theorem.

Keywords

Abelian varietiestorsion pointsnumber fieldsGalois groupsHilbertian fieldsHaran's diamond theorem

MSC classification

Secondary: 12E30: Field arithmetic

Information

Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 9 , Issue 3 , July 2010 , pp. 477 - 480
Copyright
Copyright © Cambridge University Press 2010

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References

1.Fried, M. D. and Jarden, M., Field arithmetic, 3rd edn (revised by M. Jarden), Ergebnisse der Mathematik (3), Volume 11 (Springer, 2008).Google Scholar
2.Serre, J.-P., Résumé des cours de 1985–1986 (Annuaire du Collège de France, Paris, 1986).Google Scholar
3.Mumford, D., Abelian varieties (Oxford University Press, 1974).Google Scholar