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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)?

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    $\begingroup$ mathoverflow.net/q/9984/30186 $\endgroup$ – Wojowu Jun 6 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 at 17:19

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