Mysterious Formulae

I’m not one of those mathematicians who is in love with abstraction for its own sake (not that there’s anything wrong with that). I can still be seduced by an explicit example, or even — quell horreur — a definite integral. When I was younger, however, those tendencies were certainly more pronounced than they are now. Still, who can fail to appreciate an identity like the following:

\(\displaystyle{e^{-2 \pi} \prod_{n=1}^{\infty} (1 – e^{-2 \pi n})^{24} = \frac{\Gamma(1/4)^{24}}{2^{24} \pi^{18}}}.\)

But man can not live on identities alone, and ultimately one’s efforts turn in other directions. So it’s always nice when the old and new words coincide, and an identity is revealed to have a deeper meaning. The formula above is a special case of the Chowla-Selberg formula, which is, possible typos in transcription aside,

$latex \displaystyle{\sum_{CM(K)} \log \left( y^{6} |\Delta(\tau)| \right) + 6h \log(4 \pi \sqrt{\Delta_K})
= 3 w_K \sum \chi(r) \log \Gamma(r/\Delta_K).}$

Here the notation is as you might guess — \(y\) is the imaginary part of \(\tau\), which is ranging over the equivalence classes of CM points for a fixed ring of integers in an imaginary quadratic field (there is presumably a version for orders as well). The existence of this identity (and a vague sense that it was related to the Kronecker limit formula) was basically all that I new about this identity, but Tonghai Yang gave a beautiful number theory seminar this week explaining the geometric ideas behind this formula, and some generalizations (the latter being the new work). So, just as in the Gross-Zagier paper on the special values of \(j\) at CM points, one now has *two* proofs of this result which complement each other, one analytic, and one geometric. (I apologize in advance for not being able to attribute all [or really any] of the ideas, Tonghai certainly mentioned many names but I never take notes and this was 5 days ago.) The first remark is that the RHS is essentially the logarithmic derivative of the corresponding Artin L-function. On the other hand, it turns out (non-obviously) that the left hand side can be related to the Faltings height(s) of the corresponding Elliptic curves with CM by \(\mathcal{O}_K\). I think this relation was discovered by Colmez in his ’93 Annals paper. The Faltings height has always been a slippery concept to me, and in fact the theory of heights in general has always struck me as being connected to the dark arts. In particular, various definitions depend on certain choices of height function, although they actually don’t depend on that choice in the end. So when actually doing a calculation, it’s always nice if you can magically produce some choice which makes calculation possible. And of course, when making a choice of function on some (tensor power of) \(\omega\) over the modular curve, what better choice is there (if one wants to control the zeros and poles) than \(\Delta\). (Tonghai mentioned another version of the formula where one instead used certain forms which are Borcherds products — of which \(\Delta\) is a highly degenerate example. I had the sense that this formulation was more generalizable to other Shimura varieties, but I never understood Borcherds products so I shall say no more.) Key difficulties in understanding generalizations of these formulas involve ruling out certain vertical components in certain arithmetic divisors on Shimura varieties, which I guess must ultimately be related to understanding the mod-p reduction of these varieties in recalcitrant characteristics (blech).

Colmez also formulated a conjectural generalization of the CS-formula, which is what Tonghai was talking about, and on which he (and now he together with his co-authors) have made some progress. The viewpoint in the talk was to re-interpret these identities in terms of arithmetic intersection numbers of arithmetic divisors on Shimura varieties. Of course, this is intimately related to the ideas of Gross-Zagier and its subsequent developments, especially in the work of Kudla, Rapoport, Brunier, Ben Howard, and Tonghai himself (and surely others… see caveat above). In light of this, one can start to see how special values of L-functions and their derivatives might appear. I can’t possibly begin to do this topic justice in a blog post, but I will at least strongly recommend watching Ben Howard talk about this at MSRI in a few weeks (Harris-fest, Tuesday Dec 2 at 11:00). I’ll be there to watch in person, but for those of you playing at home, the video will certainly be posted online. Ben is talking about exactly this problem. Since he is an excellent lecturer, I can safely promise this will be a great talk.

Added: Dick Gross emailed me the following (which also gives me the chance to say that Tonghai did indeed mention Greg Anderson during his talk):

************

…if you want to read a nice analytic treatment of the Chowla-Selberg formula, using Kronecker’s first limit formula, you can find it in the last chapter of Weil’s book “Eisenstein and Kronecker”.

I found an algebraic proof of C-S when I was a graduate student, using the moduli of abelian varieties with multiplication by an imaginary quadratic field (what we would now call unitary Shimura varieties). Deligne figured out what I was actually doing, and generalized it to prove his wonderful theorem that Hodge cycles on abelian varieties are absolutely Hodge.

Greg Anderson formulated a generalization of C-S for the periods of abelian varieties with complex multiplication. This was refined by Colmez, and we know how to prove all the refinements when the CM field is abelian over Q. Tonghai and Ben have been making progress in some non-abelian cases.

Dick