# partition functions and Galois representations?

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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Great! Thanks a lot! –  Thomas Riepe Jan 23 '11 at 15:30
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). –  Ramsey Jan 23 '11 at 16:16
Dear Nick, Thanks for the clarification; I'll make an edit to reflect this. –  Emerton Jan 23 '11 at 17:10