What are the nicest proofs of Jacobi’s four-square theorem you know?
(Constraint: I would like something that can be presented in two hours to a group of highly motivated students with a reasonably solid background in complex analysis but little background in number theory. Introducing key concepts (modular forms, elliptic functions) is a big plus – thus what I want is not necessarily the very shortest proof.)
I know of essentially three aproaches.
* Modular forms, as in, say, Zagier’s notes in “The 1-2-3 of Modular Forms”. The key facts here are that (a) the theta function is a modular form, (b) the space of modular forms of given weight and level is finite-dimensional (indeed 2-dimensional for the small level that is involved here). I feel reasonably certain that I can pare this down to fit my constraints without too much loss.
* Elliptic functions. This, I understand, is the route originally chosen by Jacobi himself; it is also the one I have seen most often. There are many variants. Most of the time one proves Jacobi’s triple product identity first (good: see the constraints above) and then derives the theorem from it in several pages of calculations involving at least one miraculous-looking identity. My concerns here are two-fold. First – all of these identities (triple-product included) really rest on the same fact, namely, the low-dimensionality of the space of elliptic functions with given multipliers (or whatever they are called). However, even when this is said, this is not really fully and directly exploited in the proofs I’ve seen. Second – while some proofs of Jacobi’s triple product identity I’ve seen are reasonably streamlined, what comes thereafter always comes through as computational and very ad-hoc. Can any of this be helped?
– A variant of this approach is Ramanujan’s proof in Hardy-Wright. It’s concise, but I’d really like to see what it really rests on. Can it be made clearer, a little longer and a little *less* elementary?
* Uspensky’s elementary proof, based on an identity of Liouville’s; there is an exposition of this in Moreno-Wagstaff. Liouville’s identity feels like a bit of magic to me – I feel there is really something about elliptic functions being snuck here, though I do not know exactly what.
What is your take on this? How are these approaches related? (I feel they have to be, in several ways. In particular, a Jacobi theta function is *both* an elliptic function and a modular form, depending on which variable you are varying.) What would be some ways of making the elliptic-function approach more streamlined and conceptual, without much in the way of calculations?
Here are some photos of Ramanujan’s house, courtesy of my recent trip to India. The house is essentially one-dimensional from front to back, with a corridor on the right. I was told it was felt at the time that, in a properly laid out house, one should be able to see all the way through from front to back when all doors were open.
Happy new arbitrary calendrical unit!