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Inspired by this question, over what fields is the Mordel conjecture known to be true?

For instance, is it true over fields of finite type (that is, fields finitely generated over their prime subfield)?

Improve this question
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    $\begingroup$ mathoverflow.net/q/9984/30186 $\endgroup$
    – Wojowu
    Jun 6, 2019 at 16:40
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    $\begingroup$ Given a curve of genus >1, you can apply Lang-Néron to its Jacobian, and then apply the theory of "unlikely intersections". For a field K of finite type the result is true thanks to Mordell-Lang, see Tzermias's survey on the Manin-Mumford conjecture. $\endgroup$
    – François Brunault
    Jun 7, 2019 at 17:19

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