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On torsion of abelian varieties over large algebraic extensions of finitely generated fields

Part of: Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  26 February 2010

Marcel Jacobson  and
Moshe Jarden
Marcel Jacobson
Affiliation:
School of Mathematical Sciences, Tel-Aviv University, Ramat-Aviv, Tel-Aviv, 69978, Israel.
Moshe Jarden
Affiliation:
School of Mathematical Sciences, Tel-Aviv University, Ramat-Aviv, Tel-Aviv, 69978, Israel.
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Extract

The following theorem is proved in [2[.

Let K be a finitely generated field over its prime field. Then for almost all e-tuples σ = (σ1, …, σe) of elements of the abstract Galois group G(K) of K and for all elliptic curves E defined over the following results hold.

MSC classification

Secondary: 14G05: Rational points
Type
Research Article
Information
Mathematika , Volume 31 , Issue 1 , June 1984 , pp. 110 - 116
Copyright
Copyright © University College London 1984

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References

1.Ershov, Ju.. Fields with a solvable theory. Soviet Mathematics Doklady, 8 (1967), 575–576.Google Scholar
2.Geyer, W.-D. and Jarden, M.. Torsion points of elliptic curves over large algebraic extensions of finitely generated fields. Israel J. Math., 31 (1978), 257–297.CrossRefGoogle Scholar
3.Halmos, P. R.. Measure Theory (D. Van Nostrand Company, Princeton, 1968).Google Scholar
4.Jarden, M.. An injective rational map of an abstract algebraic variety into itself. J. für reine angew. Math., 265 (1974), 23–30.Google Scholar
5.Jarden, M.. Roots of unity over large algebraic fields. Math. Annalen, 213 (1975), 109–127.CrossRefGoogle Scholar
6.Jarden, M. and Kiehne, U.. The elementary theory of algebraic fields of finite corank. Inventiones mathematicae, 30 (1975), 275–294.CrossRefGoogle Scholar
7.Lang, S.. Abelian Varieties (Interscience Publishers, New York, 1959).Google Scholar
8.Lang, S.. Diophantine Geometry (Interscience Publishers, New York, 1962).Google Scholar
9.Mumford, D.. Abelian Varieties (Tata Institute of Fundamental Research, Bombay, 1974).Google Scholar
10.Ribes, L.. Introduction to Profinite Groups and Galois Cohomology (Queen's University, Kingston, 1970).Google Scholar
11.Serre, J.-P.. Abelian l-adic representations and elliptic curves (W. A. Benjamin, New York, 1968).Google Scholar
12.Serre, J.-P. and Tate, J.. Good reduction of abelian varieties. Annals Math., 68 (1968), 492–517.Google Scholar
13.Shimura, G. and Taniyama, Y.. Complex multiplication of abelian varieties and its applications to Number Theory. Publ. Math. Society Japan, 6 (1961).Google Scholar