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• CommentRowNumber1.
• CommentAuthorTodd_Trimble
• CommentTimeMar 22nd 2014

Stubby beginning for Moebius transformation.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeMar 22nd 2014

Thanks.

• CommentRowNumber3.
• CommentAuthorTodd_Trimble
• CommentTimeMar 22nd 2014
• (edited Mar 22nd 2014)

Thanks, Urs. I added some more material, but I must have made some subtle syntax error, because for some reason it’s not rendering as it should. Perhaps I’m tired.

Edit: fixed. I still have no idea what I did wrong, but I did somehow fix whatever it was.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeApr 5th 2014

I made modular group redirect to Möbius transformation and added some more cross-links (also with moduli stack of elliptic curves etc.) This used to redirect to modular theory.

• CommentRowNumber5.
• CommentAuthorTodd_Trimble
• CommentTimeApr 5th 2014

I added material on the modular group to Moebius transformation, but I see now there’s plenty of overlap with moduli stack of elliptic curves. That’s okay I guess.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeApr 5th 2014

Thanks!!

(There is currently some syntax error that confuses the parser in the section “Action on hyperbolic space”. I would try to fix it, but I have to dash off right now…)

• CommentRowNumber7.
• CommentAuthorDavidRoberts
• CommentTimeApr 6th 2014

Can we generalise from fields to, say, integral domains? Since at the beginning of the article $k$ is a field, then soon after we have $k=\mathbb{Z}$.

• CommentRowNumber8.
• CommentAuthorTodd_Trimble
• CommentTimeApr 6th 2014

I think we can generalize to commutative rings. When it was first written it was for fields, and then later Urs began adding a bit on the modular group, which caused consideration of $k = \mathbb{Z}$. It shouldn’t surprise me a bit of people consider these things for adeles as well, which aren’t even integral domains.

• CommentRowNumber9.
• CommentAuthorDavidRoberts
• CommentTimeApr 7th 2014

I restricted to integral domains because I was wondering for what rings it makes sense to talk about the projective line. Clearly $PSL_2(R)$ makes perfect sense for any ring $R$.

• CommentRowNumber10.
• CommentAuthorTodd_Trimble
• CommentTimeApr 7th 2014

It’s a good question. We have an entry projective space which gives a general definition, but there’s also a scheme-theoretic notion of projective line (hence a functor $CRing \to Set$) and it’s not immediately obvious to me what the relation is between these notions. I suppose there ought to be on general principle a Klein-geometry approach as well, where we are first given a concrete group of transformations, and we extract a geometry from what it is the group action preserves (e.g., it could be cross-ratio or something like it).