The recent answer's link to Ono's work makes me ask and wonder if his new results on partition functions tell something about Galois representations? (Hoping that question is not a case of this)
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asked Jan 23, 2011 at 9:35
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Dear Thomas,
As far as I know, this work is not related directly to Galois representations, but is rather a particular calculation in the theory of $p$-adic modular forms (although it is not really described this way explicitly in the paper). The $p$-adic theory of modular forms of half-integral weight was developed in the 2004 Ph.D. thesis and subsequent papers of Nick Ramsey. (See Nick's comment below this answer, and the several papers available on his web-page.) The deduction of the results of Folsom--Kent--Ono from Ramsey's thesis is explained in a short note recently written by my colleague Frank Calegari.
The key idea is that iterating the $U_{\ell^2}$ operator on a space of ($\ell$-adic) modular forms of half-integral weight projects to the ordinary part of the space, which is finite-dimensional and more-or-less explicitly computable. Applying this procedure to the modular form $1/\eta$ of weight $-1/2$ (recall that $1/\eta = q^{-1/24}\prod_{n=1}^{\infty}(1-q^n)^{-1}$ is the generating function for partitions) gives the results of Folsom--Kent--Ono.
[Added: The shift from $p$-adic to $\ell$-adic in the second paragraph is made just because in the work of Folsom--Kent--Ono, and so also in Calegari's note, the distinguished prime is called $\ell$. On the other hand, when talking about this area in general, people normally speak of $p$-adic modular forms rather than $\ell$-adic ones.]
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8$\begingroup$ Actually, my thesis only treated forms of positive classical half-integral weights (those of the form $k/2$ where $k$ is odd and positive), though I doubt there is any real obstruction to using the arguments there with $k=-1$. (Also, the published article "Geometric and $p$-adic modular forms of half-integral weight" is a better reference to the content of my thesis.) However, the subsequent paper "The half-integral weight eigencurve" treats general "$p$-adic half-integral weight," which includes negative classical weights (and a lot more). $\endgroup$– RamseyCommented Jan 23, 2011 at 16:16
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$\begingroup$ Dear Nick, Thanks for the clarification; I'll make an edit to reflect this. $\endgroup$– EmertonCommented Jan 23, 2011 at 17:10