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TORSION OF ABELIAN VARIETIES OVER LARGE ALGEBRAIC EXTENSIONS OF $\mathbb{Q}$

Published online by Cambridge University Press:  10 November 2017

MOSHE JARDEN
Affiliation:
School of Mathematics, Tel Aviv University, Ramat Aviv, Tel Aviv 6139001, Israel email jarden@post.tau.ac.il
SEBASTIAN PETERSEN
Affiliation:
Universität Kassel, Wilhelmshöher Allee 71–73, 34121 Kassel, Germany email petersen@mathematik.uni-kassel.de

Abstract

Let $K$ be a finitely generated extension of $\mathbb{Q}$ , and let $A$ be a nonzero abelian variety over $K$ . Let $\tilde{K}$ be the algebraic closure of $K$ , and let $\text{Gal}(K)=\text{Gal}(\tilde{K}/K)$ be the absolute Galois group of $K$ equipped with its Haar measure. For each $\unicode[STIX]{x1D70E}\in \text{Gal}(K)$ , let $\tilde{K}(\unicode[STIX]{x1D70E})$ be the fixed field of $\unicode[STIX]{x1D70E}$ in $\tilde{K}$ . We prove that for almost all $\unicode[STIX]{x1D70E}\in \text{Gal}(K)$ , there exist infinitely many prime numbers $l$ such that $A$ has a nonzero $\tilde{K}(\unicode[STIX]{x1D70E})$ -rational point of order $l$ . This completes the proof of a conjecture of Geyer–Jarden from 1978 in characteristic 0.

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Copyright
© 2017 Foundation Nagoya Mathematical Journal  

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TORSION OF ABELIAN VARIETIES OVER LARGE ALGEBRAIC EXTENSIONS OF $\mathbb{Q}$
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