Talk:Modular form

message: We need an article on fundamental pair of periods that reviews all of the properties of a 2D lattice so that this article and the elliptic function article (and the Jacobi & Wierestrass elliptic articles) can reference it.  linas 05:10, 13 Feb 2005 (UTC)
replies: text: I think that would be excessive. A fundamental pair of periods is just an ordered pair of complex numbers that are linearly independent over the real numbers, and so form a basis for C as vector space over R. Charles Matthews 08:15, 13 Feb 2005 (UTC)
replies: text: Well, yes, but, ... as currently written, the intro isn't accessible to anyone who doesn't already know what a lattice is.  My philosophy is that the article should be accessible to a diligent undergrad or an adult reader of "average mathematical intelligence" (e.g. professors working in other unrelated fields).  An article that defines a lattice and explains that SL(2,Z) is a natural symmetry of the generator vectors would go along ways at making this article accessible. linas 17:29, 6 Mar 2005 (UTC)

text: I have added material to make the section more explicit. Charles Matthews 18:05, 6 Mar 2005 (UTC)
replies: text: Thanks! ... I haven't read it yet, but after reading it, I'm sure I will be sorely tempted to move it to its own article, since I suspect that the defintion is cluttering this page (which is why you didn't put it there in the first place). linas 18:48, 6 Mar 2005 (UTC)
thread_title: Lattice

message: On complex multiplication it is written that modular forms would explain the 
exp( pi sqrt(163) ) thing, but I can't find here any hint about how. Please elucidate me. — MFH: Talk 00:18, 22 Jun 2005 (UTC)
replies: text: I think the book by Borwein and Borwein "Pi and the AGM" covers this topic. linas 00:58, 23 Jun 2005 (UTC)

text: I've fixed this on Heegner number, which now explains it. Nbarth 19:04, 20 October 2007 (UTC)
thread_title:  complex multiplication 

message: "In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition."
Any function satisfies a functional equation and some growth condition. This introductory sentence is not very good.  —Preceding unsigned comment added by 132.206.33.88 (talk) 22:17, 27 February 2008 (UTC)
thread_title: Meaningless first sentence

message: It would be nice if this article explained to a greater extent how they can be used and what they do.. I still don't really understand.  —Preceding unsigned comment added by 89.243.76.147 (talk) 20:17, 13 April 2009 (UTC)
thread_title: Why are they interesting/how do they effect things?

message: It says it will be explained later in the article, but it isn't. Can someone who understands this please add it? Dhirsbrunner (talk) 17:11, 9 July 2013 (UTC)
thread_title: "Holomorphic at the cusp"

message: I really don't like to see things like this:
"A celebrated conjecture of Ramanujan asserted that the qp coefficient for any prime p has absolute value ≤ 2p. This was settled by Pierre Deligne as a result of his work on the Weil conjectures."
Nothing in this article mentions that Deligne "settled" the conjecture in the affirmative.   Or does someone erroneously believe the word "settled" means "settled in the affirmative"?  It does not.
Would it really have been that hard to add those three words, enlightening the mystified reader as to what Wikipedia is trying to say?Daqu (talk) 05:14, 16 August 2015 (UTC)
thread_title:  Are we trying to make readers play guessing games??? 

message: Lambda isn't a number, it's a vector. So its (-k)-th power isn't a number either, and the sum doesn't mean anything. Instead of lambda, what's needed is, depending on convention, either the length of lambda, or the square of that. Getthebasin (talk) 01:56, 6 March 2019 (UTC)
replies: text: 2-dimensional euclidean lattices can be viewed as discrete subgroups of \mathbb C, as noted in the paragraph preceding the definition. So the series makes sense. jraimbau (talk) 16:04, 6 March 2019 (UTC)
thread_title:  Eisenstein series definition needs to be fixed. 

message: I am somewhat concerned that our article works only with trivial character throughout, allowing for the presence of automorphic factors at bottom billing in the generalisations section. As a result, the (very significant) work of Deligne and Serre associating l-adic representations with modular forms cannot be understood. The assumption also means that some statements which are not quite true fly under the radar. The article claims you can ascertain dimensions of modular form spaces via the Riemann-Roch theorem. It doesn't mention that this is not true for weight 1, but it gets away with this because if the character is trivial the dimension of any weight 1 space is 0. Treating modular forms in full generality from the start is definitely not the answer. Does anyone have other ideas how this issue might be addressed?  — Preceding unsigned comment added by Awoma (talk • contribs) 09:33, 22 December 2020 (UTC)
thread_title:  Trivial nebentypus assumption 

message: In the definition of modular form, item 1 says: For any \gamma \in \Gamma there is the equality f(\gamma(z)) = (cz + d)^kf(z), and then item 2. says: For any \gamma \in \text{SL}_2(\mathbb{Z}) the function (cz + d)^{-k}f(\gamma(z)) is bounded for \text{im}(z) \to \infty.
But according to item 1, (cz + d)^{-k}f(\gamma(z)) is just f(z), so why state it this way? Why can we not say f(z) is bounded for \text{im}(z) \to \infty? Thanks for any clarification. Vda47 (talk) 15:19, 27 September 2021 (UTC)
thread_title:  Growth condition 

message: Please excuse a non-expert trying to make sense out of this. Under "Definition", it is stated that "given a subgroup \Gamma \subset \text{SL}_2(\mathbb{Z}) of finite index, called an arithmetic group, a modular form of level \Gamma and weight k is"... Hang on. \Gamma is obviously a subgroup. The level is an integer number, right? A group can't be a number (at least not as a general rule). Is there some kind of shorthand being employed here?
Wdanbae (talk) 14:16, 7 November 2023 (UTC)
replies: text: When \Gamma is a congruence subgroup it has a well-defined level which is an integer (the largest n such that \Gamma contains the principal congruence subgroup of level n). However there are many congruence subgroups with the same level so it may be clearer to use the subgroup itself as the level to disambiguate. On the other hand, in context it is often clear which family of congruence subgroups is used (usually the Hecke groups \Gamma_0) and in this case using only an integer is more convenient. This is explicitly said in the the article : https://en.wikipedia.org/wiki/Modular_form#Definition_2. jraimbau (talk) 17:28, 7 November 2023 (UTC)
thread_title:  Level = group?