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The field of rational numbers is not such a number field. That is, there does not exist a smooth projective morphism $X\to\text{Spec } \mathbf{Z}$ such that the generic fibre is a curve of genus $\geq 1$.

Which number fields allow (or do not allow) the existence of such curves?

For any number field $K/\mathbf{Q}$ of degree $>1$, does there exist a smooth projective geometrically connected curve $X$ over $K$ with good reduction over $K$?

  • 1
    "For any number $K/\bf Q$ of positive degree" should be "For any number field $K/\bf Q$ of degree $>1$" (number field, not number; and the degree $[K:{\bf Q}]$ is automatically positive).2013-10-01
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    See https://mathoverflow.net/questions/139774/2018-12-02
  • 0
    Possibly relevant: "GENUS TWO CURVES WITH EVERYWHERE TWISTED GOOD REDUCTION" available at https://projecteuclid.org/download/pdf_1/euclid.rmjm/1370267178. According to [this Master thesis](https://www.math.leidenuniv.nl/scripties/MasterSerra.pdf), « A possibly easier, but to the author’s knowledge still open, problem is the following: fix an integer $g >2$ and find an explicit example of a number field $K$ and a curve of genus g over K with good reduction at all places of $K$ »2018-12-03
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    (The MathOverflow question I quoted above only deals with potential good reduction, while in our case, $K$ is fixed at the beginning ; so it doesn't address your question. However, this MO question seems to be related to the open problem quoted above).2018-12-03
  • 0
    Related: https://mathoverflow.net/questions/316872/2018-12-04

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