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!

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.

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

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)

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

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.

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.

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

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.

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

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

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.

Goodnight.

night

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

Feel free to suggest improvements

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.

Yay @TylerLawson is here!

This is becoming a seriously badass chatroom.

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...

testing the software for the transcipt that divides the day into sections of variable lengths, as far as I know always a whole number of hours

@jimstasheff Are you able to join now?

@ManishEarth did you remove my question above?

@skullpatrol Yes. Don't spam chat with it. You got flagged.

oh

This is the homotopy theory chat not the chit chat

By the way does someone know whether it is possible to show sylow theorems using yonedas lemma ?

I have no idea.

Anybody here into spectra? I'm into spectra. Like, really into spectra.

You might want to see a doctor about that.

What are you anyway? A number theorist?

No, I prefer letters.

Yeah me too. From the look of your website, you're a category guy.

oh gee

Interesting, the notion you mentioned of connecting Lie 2-algebras to tensor categories

letters like a,b,c, not like Dear Mr ... :D

:D

@skullpatrol Let me amend that. This chatroom is not for promoting Math.SE questions

Those guys are regulars, let them chit chat

np

I assure you, I'm quite regular.

=S

I eat lots of prunes.

(anxiety is growwwwing)

this is like when my parents fight

but Manish does make a good point. we really should try to talk about homotopy theory in here

or at least.... math

because if we don't, it's just going to end up being (by the looks of it), myself, Evan, Will and maybe one or two other people

tho i'm pretty excited about the $Aut(\mathcal{Ell})$ convo that went on in here last night. that's what this room is for :))

@JonBeardsley Yeah, you should, but it's OK if you keep this as an informal chat too

What is $\mathcal{Ell}$? I just know $\ell^2$ and so

yeah. but you're right, I mean, we don't want this to be a place where people come in and say "HEY GUYS I MADE THIS COOL QUESTION ON MATH.SE CHECK IT OUT"

or even on mathoverflow

because that would be even more detrimental to the purpose of this room

A room for anyone interested in homotopy theory, or any nearby fields (e.g. category theory, algebraic geometry).

oh well. sure. i guess i wasn't thinking of that as the purpose of this room

but i see your point. i will think on how to make the "purpose" of this room more clear, though for info you can check out my answers to meta.mathoverflow.net/questions/221/homotopy-chat-room

Was anyone surprised when he heared the first time that $|\beta(\mathbb{N})|\geq |[0,1]^{[0,1]}|$ where $|\cdot|$ means the cardinality and $\beta$ is the Stone Chech compactifiction

i never heard that.

i have no feel for what stone-cech compactification does

or whether it lives inside homotopy theory, really

however it seems interesting. my advisor and i often spoke about the bousfield lattice of spectra, and there's all this crazy stuff floating around, and he would always say "well maybe it's like the compactification of the integers, you know, all kinds of uninteresting junk floating around"

at least, i think that's what he said

Well at least I shall proof this as an exercise for my topology lecture so I hope it is true :D

oh cool

@Eric you think I'll be able to work out what the fiber of $X(n)\to X(n+1)$ is by carefully building everything from the ground up?

i.e. starting with the map of spaces and super carefully examining what Thomification does?

@Eric I think it doesn't live there but I just tried to start with the math

@JonBeardsley i sort of doubt it. maybe you'll be able to work out, like, the mod-2 homology of some of the fibers for very low n

hrmmmmm yeah

well actually, so, that shouldn't be hard right? the homology of these things is completely known

that in turn may mean that you'll learn little :)

haha oh well

i'm gonna put that one on the back burner. i wanna talk to you about that cogroup scheme stuff that i was blathering about earlier, are u free right now to chat?

no serious chats today, i'm giving a talk at 2:45pm pacific

too busy being nervous

ohhhhhh that's right

:)

good luck! have you worked out that paper of Mahowald's?

yeah, the part of it i wanted to at least

you can see the present state of the notes here: math.berkeley.edu/~ericp/latex/msri-2013.pdf

oh interesting, i might give this a read

please do

though comments will be sort of moot 3 hours from now :)

hahah

no no, i'm just thinking about Thom spectra right now

so anything on them is helpful. I'm just going to have to probably sit down and get familiar with Rudyak's book at some point

i heard mike hill mention something earlier today that bob stong wrote a book on formal group laws because he didn't like any of the available treatments, and what he ended up with was a really rich description of the connection to bordism, since he's such an expert

and then he didn't publish it. and then he passed away in the late 2000s.

it would be neat to get a chance to look at the manuscript

@Jon Isn't the cardinality of $\beta(\mathbb{N})$ exactly $2^{2^\mathbb{N}}$?

@MichaelGreinecker probably you tried to ping me. I shall only show it is at least that

that sounds very interesting

i've separately heard other people complain about how poorly understood quillen's original proof of the MU_* theorem is (relative to the 'elementary proofs...' paper)

at the very least, i'm sure stong elucidated that thing

yeah, Jack says that

in particular, Jack seems to think JF Adams never really understood it

but i don't klonw

*know

@Dominic There is a MSE post about the issue.

some people in bonn were doing their best to make sense of it a while ago, but i'm not sure they managed

@ERic you know that (according to Neil Strickland at least, I haven't seen a proof of it) Thomification commutes with homotopy p.o.'s?

idk, maybe i knew that. neil has a preprint where he identifies thomification as participating in some kind of adjunction

so if that's true then it commutes with some sort of co/limits

right

i've seen various single sentences to this effect in various papers, like that Po Hu paper i mentioned at Talbot, which I have not been able to find again

which is why I'm thinking i need to just get down to nuts and bolts and work out very carefully what Thomification of morphisms looks like, at least for complex vb's

anyway. yeah, sorry, not trying to trick you into a serious convo

:)

this might mean that Richter and Baker's $M\Xi$ might be more approachable

and it's coh. surjects onto $MU$'s

oh actually scratch that. the fact that we take loops on everything ruins it

lol. no way around it, as usual

dude... the Thom iso is just a fricking shearing map

yeah. so i think your notes are going to be very helpful w.r.t my MO question on ring structures on Thom spectra

good!

@JonBeardsley, sent you an email at addresses on C.M.L. Also editted David's question a little, which now seems to have been unnecessary, plus it makes my name show up on the "active" sort. I meant well.

@Eric ECP gets nervous? Dude, your Talbot talk was baller.

oh hai sean

and hey @WillJagy

thanks for your email

just read it. that's helpful. we'll maybe consider establishing some kind of ground rules or something

maybe i'll write something and pin it

hey @Sean you there?

Yeah, trying to catch up

erm @Seantilson i guess is what i mean

oh

they both worked

@JonBeardsley, good. The thing I did not expect is that anyone on any SE site/chat can probably also post in this one.

I like @WillJagy, he seems to be a really swell guy. Helping out the youngsters and all. Thanks again Will!

(By youngsters I mean graduate students)

yeah. we need a lot of halp.

@SeanTilson, thanks. I have a few images for chat rooms, I think this one is taking a good direction already.

It really helps that people like Tyler Lawson are willing to help out the younger generation. I would name other names, but it would take too long. Last night was a great example of what this should be.

Evidently you are getting some interest from more senior people in or near the field, although they may not have figured out how to post quite yet.

Right, and Greg Stevenson peeked in a day or two ago, no post.

yeah i saw him

Jim Stasheff is trying to get in here

i dunno if he worked that out yet

Well, onward and upward. Had a root canal yesterday, it went well but i still feel a little ill. Bye for the moment.

Maybe he is having a problem similar to mine.

Bye @will

yeah, i really really love the homotopy theory community. I have no idea what other math communities are like, but this one is amazing.

like, i'm so happy i ended up being a part of it, even if it's just to meet cool people.

good luck @Eric, even though you don't need it.