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Jun 22, 2022 at 8:14 history edited CommunityBot
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Feb 4, 2010 at 6:07 comment added Pete L. Clark @TL: good question! (Sounds familiar, in fact.) Off the top of my head, I would say that the key issue is which of these algorithms work over a field of positive characteristic, and for which characteristics? E.g. "my" algorithm (a.k.a.: the most immediately obvious algorithm) for $X_1(N)$ will work verbatim over a field of characteristic not dividing $N$, hence (I'm pretty sure) over $\mathbb{Z}[\frac{1}{N}]$. Doing things correctly at primes of bad reduction will be much harder in general, I fear, although in some very special cases (e.g. genus 0!) you can work these things out.
Feb 4, 2010 at 4:44 comment added Tyler Lawson Really nice answer. One question: Do any of the algorithmic methods you mention for computing defining equations work integrally, or only over Q?
Feb 4, 2010 at 4:10 history edited Pete L. Clark CC BY-SA 2.5
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Feb 3, 2010 at 23:02 vote accept David Zureick-Brown
Feb 3, 2010 at 22:44 history answered Pete L. Clark CC BY-SA 2.5