Talk:Modular form

Lattice

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)

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)

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)

I have added material to make the section more explicit. Charles Matthews 18:05, 6 Mar 2005 (UTC)

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)

Well, no, there is no need for that. There might be a reason to add to the lattice (group) article material spelling this out for the n-dimensional case. Charles Matthews 19:16, 6 Mar 2005 (UTC)

You might want to add that modular forms need not be defined as complex analytic functions on the upper halfplane. Once you use the algebro-geomtric definition, you can consider sections of the analogous line bundles over modular curves that are defined over more general rings, for example over p-adic numbers, fields of positive characteristic, or over the integers if you invert some things. For details, see Katz' paper in the Antwerp volumes. For example, there's a lot of work done on overconvergent p-adic modular forms these days. But the reason that modular forms enter into number theory at all, in some sense, is exactly because modular curves can be defined over schemes other than Spec C. With the exception of some analytic number theory, most number theorists working with modular forms do not define them solely as analytic functions over C. They either use the algebro-geometric definition or they work with certain quotients of GL_2 over the adeles, for which their manifestation as complex analytic functions is only relevent at the infinite primes. I understand you want to keep it simple and accessible! So I don't want to (and don't know how to) edit the page. But maybe just mention other ground rings? Because, as it stands, the definition isn't the one used by many (probably most?) mathematicians who work with modular forms. (Maybe this is "POV"? Ha ha. I might be biased since I work with modular forms almost everywhere but C.) Thanks!

complex multiplication

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)

I think the book by Borwein and Borwein "Pi and the AGM" covers this topic. linas 00:58, 23 Jun 2005 (UTC)

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

Meaningless first sentence

"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)

Why are they interesting/how do they effect things?

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)

Better definition needed

In the article, condition two for a modular form f of weight k is expressed as follows:

(*)    f((az + b)/(cz + d)) = (cz+d)^k f(z)

for all (az+b)/(cz+d) such that the 2x2 matrix ((a, b), (c, d)) is a member of SL₂(Z).

But cz+d is not well-defined in terms of the fractional linear transformation T(z) = (az+b)/(cz+d); it is defined only up to sign. (T'(z) = 1/(cz+d)².) Hence for odd k, equation (*) above is not, a priori, well-defined.

So a more fine-tuned definition would be better. By someone knowledgeable in this subject, which excludes me.70.231.249.36 (talk) 22:25, 18 December 2011 (UTC)

It's true that the fractional linear transformation T(z) does not determine the matrix g = ((a,b),(c,d)) uniquely, but that is not relevant here. The quantity cz+d is perfectly well-defined, since the matrix g is an element of SL_2(Z). I think the confusion arises because of the unnecessary "for all (az+b)/(cz+d)".

I added a comment on this right after the definition, I used the comment after Definition 1 in [1] as a reference. Evilbu (talk) 14:46, 19 July 2013 (UTC)

Evilbu (or whoever wrote it; I can't tell, thanks to the lack of signatures or maybe the poor formatting), this quote:

"It's true that the fractional linear transformation T(z) does not determine the matrix g = ((a,b),(c,d)) uniquely, but that is not relevant here. The quantity cz+d is perfectly well-defined, since the matrix g is an element of SL_2(Z)"

from your post makes no sense to me. Can you please explain exactly how "The quantity cz+d is perfectly well-defined"? Because as far as I can tell, it is only defined up to sign, which would contradict the well-definedness of the condition (*) above.

The map from SL₂(Z) to linear fractional transformations {(az+b)/(cz+d)} (with a, b, c, d integers such that ad-bc = 1) is two-to-one, with the inverse image of any specific (az+b)/(cz+d) being two matrices in SL₂(Z) that differ by the signs of all four integers a, b, c, d. This means that, given (az+b)/(cz+d), the expression cz+d is defined only up to sign.Daqu (talk) 05:33, 16 August 2015 (UTC)

But you know the sign of c and d, so what's the problem, exactly? 67.198.37.16 (talk) 23:09, 25 January 2019 (UTC)

"Holomorphic at the cusp"

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)

Elliptic curve

The part "Definition_in_terms_of_lattices_or_elliptic_curves" doesn't refer to elliptical curves at all, at least not recognizably so. unsigned

Are we trying to make readers play guessing games???

I really don't like to see things like this:

"A celebrated conjecture of Ramanujan asserted that the q^p coefficient for any prime p has absolute value ≤ 2p<sup?11/2. 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)

Eisenstein series definition needs to be fixed.

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)

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)

Oh, you're right of course. I missed "Two-dimensional". Carry on! Getthebasin (talk) 19:48, 7 March 2019 (UTC)

Trivial nebentypus assumption

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)

Growth condition

In the definition of modular form, item 1 says: For any $\gamma \in \Gamma$ there is the equality $f(\gamma (z))=(cz+d)^{k}f(z)$, 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)

Level = group?

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)

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)