The recent [answer's](https://mathoverflow.net/questions/52744/what-is-the-relationship-between-modular-forms-and-maass-forms/52753#52753 "link") link to Ono's work makes me ask and wonder if his [new results](http://www.aimath.org/news/partition/folsom-kent-ono.pdf "pdf") on partition functions tell something about Galois representations? (Hoping that question is not a case of [this](http://m.discovermagazine.com/2010/sep/25-modern-humans-smart-why-brain-shrinking/article_view?b_start:int=0&-C= "link")) 



partition functions and Galois representations?
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](http://condor.depaul.edu/nramsey/math/papers/harvard_thesis/huthesis.pdf) and subsequent papers of [Nick Ramsey](http://condor.depaul.edu/nramsey). (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](http://www.math.northwestern.edu/~fcale/Files/FKO.pdf) recently written by my colleague [Frank Calegari](http://www.math.northwestern.edu/~fcale).


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.]


