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Optimal quotients and surjections of Mordell-Weil groups. (English) Zbl 1417.11113
Summary: Answering a question of Ed Schaefer, we show that if \(J\) is the Jacobian of a curve \(C\) over a number field, if \(s\) is an automorphism of \(J\) coming from an automorphism of \(C\), and if \(u\) lies in \(\mathbb{Z} [s] \subseteq \operatorname{End} J\) and has connected kernel, then it is not necessarily the case that \(u\) gives a surjective map from the Mordell-Weil group of \(J\) to the Mordell-Weil group of its image.
MSC:
11G10 Abelian varieties of dimension \(> 1\)
11G05 Elliptic curves over global fields
11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
11G35 Varieties over global fields
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