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On the rank of abelian varieties over ample fields. (English) Zbl 1193.14054

A field \(F\) is said to be ample (or large) if any smooth curve over \(F\) has either infinitely many \(F\)-points or none. For example, an algebraically closed field is ample. It is conjectured here that if \(F\) is an ample field that is not algebraic over a finite field then any nontrivial abelian variety \(A\) over \(F\) has infinite Mordell-Weil rank. Note that if \(F\) is algebraic over a finite field then every abelian variety over \(F\) has rank zero.
The main result here is that under these conditions \(A(F)\otimes{\mathbb Z}_{(p)}\) is not finitely generated as a \({\mathbb Z}_{(p)}\)-module if \(F\) has characteristic \(p\geq 0\). In particular the conjecture holds if \(F\) has characteristic zero. The proof uses the Mordell-Lang conjecture, established by Faltings and others. The weaker statement that \(A(F)\) is not finitely generated is given an alternative, elementary, proof. This is then used to improve on some earlier results by showing that if \(K\) is an infinite finitely generated field and \(A\) is defined over \(K\), then \(A\) has infinite rank over a large class of ample extensions of \(K\).

MSC:

14K05 Algebraic theory of abelian varieties
12E30 Field arithmetic
14G05 Rational points
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References:

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