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I am giving an extremely short talk (~12 minutes) in a few weeks in which I need to be able to introduce and motivate the idea of a classical modular form to an audience of undergraduates in as short a time as possible. I was intending to introduce modular forms as functions on lattices, following Serre's presentation in A Course in Arithmetic - this seems like the shortest path to motivating the particular symmetries that we require of modular forms. Does anyone else have any good ideas? Particularly, if there is a way of motivating forms of higher levels and of half-integral weight through the same idea of a function on lattices, I'd like to hear about it.

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I like Noam Elkies' presentation in <a href="ams.org/notices/200011/fea-elkies-2.pdf">Lattices, Linear Codes, and Invariants</a>. –  Qiaochu Yuan Nov 19 '09 at 21:20
    
Link is broken - should be ams.org/notices/200011/fea-elkies-2.pdf –  Nick Salter Nov 19 '09 at 22:56
    
If you write some PDF for the talk (maybe after finishing it), I would be happy to read it and know if it "worked". –  Konrad Voelkel Nov 20 '09 at 8:51
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3 Answers 3

12 minutes??! Jeez. I might just say "modular forms are to Mobius transformations as trigonometric functions are to translations." For higher level point out that sin( pi x/N) transforms nicely under a smaller group than does sin(pi x). For half-integer weight I have no idea.

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Thanks; I was thinking of introducing them by analogy with the trigonometric functions. The only issue is that with trig functions, their "automorphy factor" is always 1 and I want a good way of motivating the particular factors that arise in modular forms. I do like the observation with sin( pi x/N). –  Nick Salter Nov 20 '09 at 4:35
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Couldn't you say that sin(pi x) transforms under Z with nontrivial factor of automorphy given by j(n) = (-1)^n? –  Tom Church Nov 20 '09 at 19:55
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There is a nice introduction to that circle of ideas (Modular Forms) using only basic complex analysis in Freitag's complex analysis book: http://www.amazon.com/reader/3540939822?encoding=UTF8&ref=sib_dp_pt#reader-link

It is intended to be for undergrads. Moreover, the last chapter the author is trying to use such modular forms and basic complex analysis to solve two very specific problems in number theory. This motivates all the machinery introduced very nicely. Surely you can find there some good ideas to talk about.

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I would suggest pointing out the trigonometric analogy (as JSE suggested) and mentioning that modular forms also have Fourier expansions, by virtue of invariance under upper-triangular matrices. Since you want to introduce weights, I also suggest including a computation of the differential d(az+b/cz+d) to motivate the appearance of the factors (cz+d)^k. This lets you describe weight k forms as k/2-fold polydifferentials or sections of a pluricanonical bundle (and are in general not differential forms, since you use tensor powers instead of wedges). Your time constraint is rather unrealistic, but modular forms of half-integral weight can be motivated by theta functions of odd-dimensional lattices. I don't know how to phrase this in the language of lattices in C.

Higher level forms can be introduced as functions on a space of pairs (or more generally, diagrams) of lattices with isogenies of a fixed type between them. I suppose it's a rather awkward perspective, but it's hard to introduce a new language in 12 minutes.

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