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a theorem of Fontaine states that there are no curves of genus $\geq 1$ over $\mathbb Q$ with everywhere good reduction.

For curves of genus one over number fields, this is not true. There are number fields where one can construct elliptic curves with everywhere good reduction (e.g. http://arxiv.org/abs/1410.0651).

But what if we go over to genus $\geq 2$ ? Is there a statement like

There is no curve of genus $\geq 2$ over a number field with everywhere good reduction.

?

Kind regards,

reinbot

Dan
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    http://mathoverflow.net/questions/139774/curves-with-good-reduction-everywhere – Daniele A May 11 '16 at 09:58
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    Related: https://math.stackexchange.com/questions/86900/ – Watson Dec 02 '18 at 22:30
  • Does the example on the top of page 4 here answer your question? – Watson Dec 03 '18 at 12:59
  • Moreover, https://mathoverflow.net/questions/139774/ tells you that for every genus $g$, there is a number field $K$ and a curve $X$ over $K$, such that $X$ has good reduction at all the places of $K$. – Watson Dec 03 '18 at 13:24
  • The link in the comment from Dec 3, 2018 at 12:59 is now https://www.universiteitleiden.nl/binaries/content/assets/science/mi/scripties/masterserra.pdf, or see https://i.sstatic.net/TOHPk.png – Watson Oct 27 '22 at 06:12

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