Talk:Automorphic form

Condition 2 isn't quite right, it seems. The analytic conditions are something like 'within a finite-dimensional space'.

Charles Matthews 21:37, 6 Sep 2004 (UTC)

Clean up messages

A tagger slapped four different maintenance tags on this article and did not explain any of them. They were:

1. clean-up jargon,
2. notability,
3. in-universe, and
4. tone

Of these, the in-universe really makes no sense, unless it was meant as some kind of philosophical joke. After that, notability makes little sense, because this is a concept which has been thoroughly studied by mathematicians. I don't see tone applying here, but clean-up jargon is the only tag which I can see as kind of making sense for this article as it stands today.

But since the tagger failed to explain any of these, we could feel perfectly justified in removing all of them. PrimeFan 23:00, 31 October 2007 (UTC)

I've gone ahead and removed the jargon tag. The exposition here is already clearer than can be found in most treatments of the subject, I believe. Considerable effort has obviously been made to making the article accessible to the lay mathematician. Sławomir Biały (talk) 14:55, 28 August 2009 (UTC)

The definitions should become more precise though. I don't think that after reading this it is possible to know what an automorphic form is: it only gives a vague idea. —Preceding unsigned comment added by 129.175.50.70 (talk) 12:33, 20 October 2009 (UTC)

Too much jargon!

I spent a term in a graduate course on automorphic forms and still didn't work out what they are. It is impossible to talk about them without jargon. —Preceding unsigned comment added by 128.40.56.75 (talk) 17:36, 13 March 2008 (UTC)

Formulation enhancements

Concerning "too much jargon", I think it's mainly the first paragraph in the section "Formulation" ("The formulation requires ... of the chain rule.") that is scaring. I'd propose to move that paragraph further below, since it's the second one ("In the general setting, then,...") that gives a first rough definition of automorphic form.

Also I would propose to enhance the first sentence of that second paragraph, to make clearer what G and Γ are. Here is a proposal (I'm not doing the edit myself since I don't know whether it's correct):

Let G be a Lie group and let Γ be a discrete subgroup of G. In this setting, an automorphic form is a function F on G with values in ℂ (?) (or in some fixed finite-dimensional vector space V, in the vector-valued case), subject to three kinds of conditions:

128.176.171.236 (talk) 13:42, 13 April 2013 (UTC)

Todo

• Add examples of automorphic forms
• Automorphic Forms on Adele Groups - Gelbart - excellent resource
• An introduction to the langlands program - ch 3
• Course notes with similar content to Gelbart's book - https://services.math.duke.edu/~jgetz/aut_reps.pdf
• http://www2.math.ou.edu/~kmartin/papers/mfs.pdf is a good reference
• http://www.math.columbia.edu/~chaoli/docs/AutomorphicForm.html too

Overview reference

https://authors.library.caltech.edu/43491/ volume III

Examples

• GL_1 from Dirichlet characters
• GL_2 from modular forms
• Rankin–Selberg convolution giving GL_4 automorphic forms (Deitmar - Automorphic Forms)

As sections of shimura varieties

• http://www.numdam.org/item/SB_1970-1971__13__123_0/
• http://www.math.columbia.edu/~harris/MathG6245.htm - Coherent Cohomology of Shimura Varieties

— Preceding unsigned comment added by Wundzer (talk • contribs) 00:42, 24 July 2020 (UTC)

Langlands program, and Implications

In modern mathematics automorphic forms and their functions play a pivotal role around unification through algebraic geometric number theory. A section needs to exist about the Langlands correspondences and its conjectures around automorphic forms, explaining how exactly their analytical nature is constructive in this field; probably by an exposition with Artin reciprocity. Also some formulas from associated papers with analytical implications (such as the connection to L-functions and the Riemmann hypothesis), exemplifying the exact and as yet unexplained on the wiki (also perhaps a bit generally) nature of automorphic functions. And, how they construct the Laglands philosophy through its fundamental lemma. Granted a specialist topic, but one that has far reaching consequences for mathemaics as a whole; and an excellent starting point for high theoretic research.

In other words, we need some geeks...

5.90.114.126 (talk) 11:52, 28 June 2021 (UTC)

Definition section is confusing

The definition section first (paragraph "In mathematics, ...") defines an "automorphic form" as a function f : X -> C satisfying a single condition (translation by elements of G with the automorphic factor).

After that, in the paragraph "An automorphic form is a function F on G", the same term "automorphic form" is suddenly, without any connecting text, defined as a function F : G -> C or F : G -> V, respectively, (so now it's a function on the group rather than on the manifold being acted on) subject to *three* conditions, of which only the first seems to connect to the previous paragraphs.

Is the first "definition" not a complete definition, or am I missing some connections here? In any case, it would help a lot if the text would explain the connection of these two different attempts at definition.

I'm also not sure whether all the rest of the section, including the examples, and "... automorphic forms can be thought of ..." should really be part of the "Definition" section. ESteiner (talk) 09:40, 14 July 2023 (UTC)