Talk:Modular form

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[edit] 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!

[edit] 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)