A short version of my question is: Is there a $p$-adic theory of integration?
Now let me expand a little further. In introductory texts such as Koblitz' book $p$-adic numbers,.. a bunch of $p$-adic analysis is developed. However, since all applications are towards number theory, the exposition stops at some point. In particular, there is no theory of integration developed for $p$-adic numbers.
By this I do not mean putting a measure on $\mathbb{C}_p$ and integrating real or complex valued functions, but instead putting a "$p$-adic measure"(whatever this may be) on it and integrating $\mathbb{C}_p$-valued functions on it.
To rephrase my question: Is there are an integration theroy for $\mathbb{C}_p$-valued functions on $\mathbb{C}_p$. In particular I would like to know if an analogue of Cauchy's theorem holds. Where can I read more about such a theory?
Best Answer
There is an important difference, relevant to the original question, between the two kinds of $p$-adic integrals mentioned by Kevin in his comments. Because I see frequent confusion on this issue, I thought I'd comment.
The 'usual' $p$-adic integrals as you might see in, say, Tate's thesis on L-functions or the adelic theory of automorphic forms, are volume integrals, with respect to a measure, typically on some group. This kind of volume integral can also be easily defined on arbitary varieties, and you will see plenty in Weil's book on Tamagawa numbers, or in papers on motivic integration. Coleman integration, on the other hand, is a $p$-adic analogue of line integrals, and comes up most naturally in discussing the holonomy of vector bundles with connection on a variety over a $p$-adic field (often interpreted as isocrytals). These, therefore, should be the right quantities to relate to a Cauchy formula. However, unfortunately (and fortunately), it doesn't work. The reason is that Coleman integration is a line integral along a canonical path between two points on a variety over the $p$-adics. So there is a canonical holonomy in the theory, at least if you just want to compute it for a bundle with unipotent connection, that is, one that has a strictly upper-triangular connection form. This is where a mysterious 'crystalline' structure on the space of paths is used, whereby there is a unique path invariant under the action of the Frobenius. The notion of a path, by the way, uses the Tannakian formalism in this context. For a very quick overview of this approach, you can look at section 2 of this paper: http://www.ucl.ac.uk/~ucahmki/siegelinv.pdf
Breuil's paper linked from Chandan's answer should provide a more systematic overview.
Anyways, because of the canonical paths in Coleman's theory, there can be no holonomy around a loop, and hence, no Cauchy formula. I was told quite a few years ago by Berkovich that he has a theory of line integrals on Berkovich spaces that are path dependent in interesting ways, but I've never looked into it.
Added: I realize I didn't mention above the connection between holonomy and usual integration of a one-form $A$. You get this by considering the connection $$d+\begin{bmatrix}0& A; \\ 0& 0\end{bmatrix}$$
on the trivial bundle of rank two. One view of Coleman integration is that the holonomy $H_a^b$ from $a$ to $b$ is defined first. And then, the naive integral is defined by the fomula $$H_a^b=\begin{bmatrix}1& \int_a^bA ;\\ 0& 1\end{bmatrix}$$