Homotopy Theory - 2013-06-28 (page 4 of 5)

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5:00 AM

Is there any reason to think that Aut(Ell) is interesting?

(That is, nontrivial?)

I mean, other than -1 as an automorphism of every elliptic curve, which I suppose would make it something like the 2-group (Z/2)[1]

If I'm understanding this correctly.

Well, no - it would be very interesting if it were trivial, but also very interesting as a natural example of a 2-group if it were not

I think we may be talking at cross purposes

Maybe

Imagine the automorphism group of the coarse moduli space of curves (let's magic away problems here), as a scheme

brb

so can I do some sanity checks here

please do!

5:04 AM

if I just have a scheme considered as a stack, what is this automorphism 2-group?

just the automorphism group as a scheme

yes ok great

So given a stack X with a coarse moduli space M, we get a map from the automorphism 2-group of X to the automorphism group of M

Does the coarse moduli space have any nontrivial automorphisms?

That's a good question.

5:07 AM

I don't think it does.

I could be wrong, though

Hm, ok

can you just pretend any such automorphism would have to be a Möbius transformation fixing 0, 1728 and infinity and so is the identity?

I don't know

And I probably want to not just work over C here

yeah ok sure

Otherwise I'm only considering the fibre of Ell over C, and perhaps there it is rigid, but over other rings there may be more automorphisms

5:12 AM

Maybe @TylerLawson knows the answer!

Well, that's the joy of MO

argh, curiosity + autosignin

Hi @TylerLawson.

We're trying to figure out whether the automorphism group of the coarse moduli space of elliptic curves is nontrivial

As a first step to asking about the autoequivalence 2-group of the stack Ell

Autoorphism group or groupoid?

First the automorphism group of the moduli space (ignoring everything but its scheme structure)

5:15 AM

The coarse moduli is P^1 over Z, so it has a PGL_2(Z) action.

Excellent!

Err, PSL_2(Z). Sorry.

So Aut(Ell) should be highly nontrivial

Automorphisms of the moduli stack are going to preserve the divisors that have points with larger automorphism groups.

Namely those with j-invariant 0 and 1728.

Yes, that is true

5:17 AM

From a practical point of view this means that you only have a very narrow possible range of automorphisms left.

So what possible automorphism groups are there for elliptic curves?

More than that, right? Elliptic curves over finite fields can have lots of nontrivial automorphism rings.

Err, endomorphism rings.

Well, there's a range of them. But the points with large Aut-groups are all defined by choise of (a prime) + (a j-invariant)

So if we only know that j=0 and j=1728 are fixed, then the primes are also fixed

and so I think that we don't gain new information there.

Ah, I see. I think.

More to the point, the set of C-points of P^1 is dense in P^1_Z, and so auts that fix the complex points also fix P^1_Z.

5:21 AM

Hmm, so there should be some interesting things to say given my putative question

So does this mean that $π_0$ of the automorphism 2-group is something like... $\mathbb{G}_m$?

This is automorphisms of $\mathbb{P}_1$ fixing two points, no?

ok. But are there auts that don't fix the C-points?

I don't think so. I'm trying to come up with an argument that infinity should be fixed, too.

I suspect the aut groupoid is probably BZ/2.

Note that Aut(coarse moduli space) is not equal to pi_0 of Aut(Ell)

@EvanJenkins - ok ok

I deleted my irrational exuberance.

5:26 AM

Hmm, perhaps I need to ask the question, unless Tyler wants to amaze everyone by posting an answer instantly after working it out here.

he doesn't need that to amaze everyone anyway

my suggestion would be: you should calculate Aut of the upper half plane quotient stack, i.e. the C-points

No, I mean *instantly*, as in prewritten and cut and paste in :-)

And now I found how to do multiline comments

Isn't $\infty$ not really part of the moduli stack anyway? Doesn't it mean that the curve is degenerate?

ah, thought you meant the compactified moduli.

5:30 AM

Well, even if it is the compactified moduli, surely the degenerate curve can't be sent to a nondegenerate one, no?

that's easier, infinity is then the unique compactfication point over C and is automatically fixed.

Yeah

evan: well, problem is that it's a natural transformation of functors

Yep.

Mmm

5:30 AM

there are certainly weird natural transformations Ell -> Ell

This is what I'm interested in

(e.g. those sending ALL points to the point with j=16)

OK, not nat transf, but natural equivalences

anyway, you can view the stack over C as the quotient of P^1(C) \ P^1(R) by GL_2(Z)

and I mean equivalences of groupoid-valued functors

5:32 AM

the aut groupoid should be calculable from that.

sorry, too tired and have to get to bed.

I think you just get Z/2.

The only generic nontrivial automorphism of an elliptic curve is -1.

That's ok :-)

Yeah.

It's just a question of path components.

Yep

'night.

5:34 AM

Goodnight.

night

Well, I'll put something together and then it can get a proper answer on MO

5:48 AM

0

Q: What is $Aut(Ell)$?

Consider the stack $Ell$ (of groupoids) of elliptic curves. I'm interested in the autoequivalence 2-group of $Ell$, the objects of which consists of transformations $Ell \Rightarrow Ell: Ring \to Gpd$ valued in equivalences of groupoids. The arrows are isomorphisms of such transformations. In a ...

ag.algebraic-geometry ct.category-theory elliptic-curves algebraic-stacks

Feel free to suggest improvements

6:01 AM

Hm, maybe you do get more than just $\mathbb{Z} / 2$, because over finite fields, you have the Frobenius map.

Oh, never mind, that's not invertible.

9 hours later…

3:02 PM

Yay @TylerLawson is here!

This is becoming a seriously badass chatroom.

3:20 PM

Was here.

yeah

but

he was actually here when i wrote that. and then he left

he probably got a ping and was like "oh man, I'm still in that chatroom?"

hmm...

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