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)
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 halfintegral 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 webpage.) The deduction of the results of FolsomKentOno 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 halfintegral weight projects to the ordinary part of the space, which is finitedimensional and moreorless explicitly computable. Applying this procedure to the modular form $1/\eta$ of weight $1/2$ (recall that $1/\eta = q^{1/24}\prod_{n=1}^{\infty}(1q^n)^{1}$ is the generating function for partitions) gives the results of FolsomKentOno. [Added: The shift from $p$adic to $\ell$adic in the second paragraph is made just because in the work of FolsomKentOno, 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.] 

