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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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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