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Fix a number field $K/\mathbf{Q}$, a finite set of places $S$ in $K$, an integer $g$ and a curve $X$ over $K$ of genus $g$.

Is there an algorithm which tells you if $X$ has good reduction outside $S$? If yes, does this algorithm run in polynomial time?

I think the answer depends on how one "gives" X. We can give $X$ by explicit equations in some projective space, as a branched cover of $\mathbf{P}^1$ or etc.

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I think that this is a great question and I'm keen to see that it does not get lost and forgotten. In order to help it along I have a few naive remarks. As I am sure you are aware, for elliptic curves we have Tate's algorithm, which given a Weierstrass equation can calculate the conductor and hence the primes of bad reduction. However, do you know if there is an algorithm for curves of genus one without rational points? Given, say, as a non-singular plane curve of degree three? –  Daniel Loughran Jan 9 '12 at 19:47

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