# curves with good reduction everywhere

It seems to be a folklore that for any 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 $K$. Are any simple proofs of this?

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Perhaps you could just construct them; my guess is that something like $y^2=x^n+1$ has potentially good reduction at every prime. –  Tim Dokchitser Aug 19 at 8:05
One way of seeing this is by appealing to Rumely's general local-global principle over $\bar{\mathbb{Z}}$, applied here to the moduli stack: an algebraic scheme over the algebraic integers $\bar{\mathbb{Z}}$ has a solution (point) in $\bar{\mathbb{Z}}$ if and only if it does in all $v$-adic completions $\bar{\mathbb{Z}}_v$. I don't know if this is a simple proof, though - it is probably not what you are looking for.
You can also apply Rumely's theorem directly to the moduli space of tricanonically embedded smooth curves of genus $g\geq2$: this is a scheme (subscheme of some Hilbert scheme), which is quasiprojective, smooth and surjective over $\mathrm{Spec}\:\mathbb{Z}$. Moreover it is a $GL_N$-bundle over the stack $M_g$ for some $N$, hence has geometrically connected fibers over $\mathrm{Spec}\:\mathbb{Z}$ by Deligne-Mumford. So Rumely's theorem applies. –  Laurent Moret-Bailly Aug 19 at 14:52