TORSION OF ABELIAN VARIETIES OVER LARGE ALGEBRAIC EXTENSIONS OF

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[STI...

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Bibliographic Details
Published in:Nagoya mathematical journal 2019-06, Vol.234, p.46-86
Main Authors: JARDEN, MOSHE, PETERSEN, SEBASTIAN
Format: Article
Language:English
Subjects:
Algebra
Eigenvalues
Geometry
Mathematics
Prime numbers
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Summary: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.
ISSN:0027-7630
2152-6842
DOI:10.1017/nmj.2017.33