MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.