The following seems curious to me:

so adelic automorphic forms are away to unify modular forms as one ranges over all level structures on elliptic curves with congruence subgroups $\Gamma_0(N)$ as $N$ ranges.

There is another context in which one sees a similar unification of all the $\Gamma_0(N)$: in modular equivariant elliptic cohomology as obtained by Hill-Lawson.

]]>The paragraph *Modular forms as adelic automorphic forms* used to jump into some discussion without any motivation. I have added at least the following lead-in paragraph:

]]>Where by the above an ordinary modular form is equivalently a suitably periodic function on $SL(2,\mathbb{R})$, one may observe that the real numbers $\mathbb{R}$ appearing as coefficients in the latter are but one of many p-adic number completions of the rational numbers. Hence it is natural to consider suitably periodic functions on $SL(2,\mathbb{Q}_p)$ of all these completions at once. This means to consider functions on $SL(2,\mathbb{A})$, for $\mathbb{A}$ the ring of adeles. These are the

adelic automorphic forms. The may be thought of as subsuming ordinary modular forms for all level structures.

Thanks for the pointer, Charles.

My impression was that “Deligne’s definition” referred to by Zoran is the one that says that automorphic forms are sections of suitable line bundles on Shimura varieties.

In the article that you point to he indeed talks about what are called (e.g. Martin 13, p. 7) “adelic automorphic forms”, namely reps of $GL_n(\mathbb{A}_F)$ on spaces of certain well-behaved functions on $GL_n(F)\backslash GL_n(\mathbb{A}_F)$.

Now both these two definitions may be related – and that was my question, how they are generally related (I know only that they are equivalent for elliptic moduli on one hand and $GL_2(\mathbb{A}_{\mathbb{Q}})$ on the other, e.g. Martin 13, p. 8).

In section 2 of the above article Deligne talks about modular forms, which should mean classical modular, i.e. sections on the moduli of complex elliptic curves. That should be related to what I mentioned in #27 I would like to see the generalization of.

(If I sound like I am missing something basic, then that’s likely because I am. Just set me straight.)

]]>By “Deligne’s definition”, do you mean what he talks about here: http://publications.ias.edu/sites/default/files/Number21.pdf. That looks like a definition of “adelic automorphic forms” to me, but perhaps I misunderstand what that is supposed to mean.

]]>But maybe I haven’t looked at a good enough discussion of Deligne’s definition yet. Which text would you recommend?

]]>What Deligne defined are complex-geometric modular-like classical-like automorphic forms, no? He didn’t consider adelic automorphic forms, did he?

That’s what I am after here, adelic automorphic forms that happen to be equivalent to those complex/modular/classical-like automorphic forms.

So by the above for the number field being just $\mathbb{Q}$ itself then the adelic automorphic forms are essentially equivalent to the standard modular forms. I suppose more generally Hilbert modular forms have an adelic automorphic incarnation (though right now I am not sure which one).

What is more generally the relation? Which of Deligne’s automorphic forms have an adelic automorphic incarnation, and of which form?

]]>It says that some of the number theoretic automorphic forms, all the way down over $Spec(\mathbb{Z})$, are again just sections on the moduli space of complex elliptic curves.

Deligne *defined* automorphic forms as something like sections of automorphic bundles over modular varieties…

I must say I had earlier not been really aware of this equivalence

$\Gamma \backslash PSL(2,\mathbb{R}) \simeq Z(\mathbb{A}) GL_2(\mathbb{Q})\backslash GL_2(\mathbb{A})/ GL_2(\mathbb{A}_{\mathbb{Z}})$Isn’t that striking? It says that some of the number theoretic automorphic forms, all the way down over $Spec(\mathbb{Z})$, are again just sections on the moduli space of complex elliptic curves.

Am I missing something, or does that not make one wonder about the whole idea of the analogy to be found here?

What’s the most general (or a more general) relation between adelic automorphic forms and classical automorphic (i.e. modular) forms?

]]>Zoran, thanks, I’ll look into that (not right now maybe, my battery is dying this moment).

Meanwhile I have expanded a bit more. Added more sentences here on how traditional modular forms on the upper half plane are automorphic functions on $PSL(2,\mathbb{R})$, and then here on how these in turn are equivalently adelic automorphic functions, due to the equivalence

$\Gamma \backslash PSL(2,\mathbb{R}) \simeq Z(\mathbb{A}) GL_2(\mathbb{Q})\backslash GL_2(\mathbb{A})/ GL_2(\mathbb{A}_{\mathbb{Z}}) \,.$Added a bunch of pointers to the notes Martin 13, which I find a nice concise collection of the relevant stuff.

]]>Siegel’s textbook (vol 2 of Complex Function Theory) and the Encyclopaedia of Mathematics, for instance, distinguish automorphic forms and a more general automorphic functions We do not have even a redirect for automorphic function. Our idea section fits more with the more general notion while the middle of the text with the form version.

http://www.encyclopediaofmath.org/index.php/Automorphic_form

http://www.encyclopediaofmath.org/index.php/Automorphic_function

]]>I have edited the Idea-paragraphs at *automorphic form* a little more in an attempt to bring out better how the concept evolved in time, for instance mentioning (and making redirect) the old term “Fuchsian functions”.

But this still has loads of room for improvement.

Good MO comments on the historical aspect of the terminology are this one and [this one] (http://mathoverflow.net/a/21556/381) (linked to now from the entry).

]]>So in each context one adds conditions that these functions on these cosets are suitably well-behaved. But what this means is not set in stone and is being adapted as necessary in applications. But I have added to the entry now some indications of some such conditions.

Regarding the homotopification: so if geometric Langlands is correct then we are done: the homotopification of automorphic functions is Hitchin connections/prequantum line bundles on moduli $\infty$-stacks of higher gauge fields.

]]>I hadn’t realised they were quite so general. So a function on a homogeneous space counts?

By the time it’s been properly homotopified, what results? Functors on certain action $\infty$-groupoids?

]]>I have tried to give *automorphic form* an Idea section with some minimum actual idea in it. Of course this needs to be expanded a lot further.

Anything on the link between height and detection of $v_n$ periodic behaviour would be good to add, e.g., Ravenel on p. 15 of these slides.

]]>Wow. I wrote up the page height of a variety over a year ago. I’m shocked that it comes up in this topic. My thesis work has a lot to do with height, p-divisible groups, and liftability for K3 surfaces and Calabi-Yau threefolds and the relation to the derived category. This is very interesting since the motivation was very different for me.

]]>Re 16, that would seem to be a good move to link your work up to topological automorphic forms. There does seem to be considerable interest.

If I have this height business correct, I think Fivebrane would pick up $v_6$ periods. It seems that K3-cohomology can get up to height 10, and the Shimura variety approach could get to any height.

]]>Thanks Todd for Nolan. It sliped my mind. Is anybody having a fie of his book on Fourier transform and symplectic geometry. It is a simple old book whose simplicity made me always easily remember the stuff. But I have not seen it since leaving United States…

]]>I have checked again with Hisham Sati. He tells me that in the 2008 talk where he talked about Fivebrane structures, he already stated a conjecture that there will be a morphism from $\Omega^{Fivebrane}$ to topological automorphic forms. I didn’t know about that, to be frank.

If I find out more that I may share, I’ll let you know.

]]>I don’t have any definite answers to these questions. It seems that most the 6-d analogs of the corresponding 2-d ingredients of the story are very much not understood yet.

]]>But what links all this to the homotopy groups of the sphere? What is the equivalent for Fivebrane of TMF for String? Is there a ring-valued genus from $\Omega^{Fivebrane}$ to some ring?

]]>Perhaps a crazy thought, but looking at this slide from a talk by Behrens, wouldn’t you expect an extension of the Whitehead tower to Fivebrane to give something interesting?

Yes, certainly, that’s why the Fivebrane group is called such: just as Spin-structures make the super-particle i.e. the super 1d QFT be well-defined, and String-structures makes the heterotic string, i.e. the super 2d QFT be well defined, so Fivebrane structures similarly relate to super 6-dimensional QFT.

But, while we know that FIvebrane structures cancel the “fermionic anomaly” of the 5-brane, otherwise very little is known about that 6d QFT, as of yet.

]]>Perhaps a crazy thought, but looking at this slide from a talk by Behrens, wouldn’t you expect an extension of the Whitehead tower to Fivebrane to give something interesting?

As you co-kill the homotopy groups, $O(n)$ comes to resemble the trivial group more closely, and we get closer to what Behrens calls $\Omega^{e}_{\ast}$, isomorphic to stable homotopy. So $\Omega^{Fivebrane}_{\ast}$ should see more $v_n$ periodic behaviour.

]]>The last two items correlate clearly. But I am not sure how to see why it is specifically TAF that comes out for higher dimensional SQFTs. If it does.

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