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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?

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wood, last semester I gave a talk on similar things. I started with Silverman's excellent book "Arithmetic of dynamical systems" and then I jump to Baker's Potential Theory on Berkovich spaces to show the existence of some kind of Lyubich measure. There are also many papers about p-adic functional analysis and they should use measure theory. –  Niyazi Mar 7 '11 at 13:24
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There are several ways to interpret this question; perhaps the questioner doesn't want to make the question more precise though, because leaving it vague may maximise the amount of answers he/she will get! There is a perfectly good Haar measure on $\mathbf{Q}_p$, and so one can do classical integration on this, and integrate $L^2$ functions and so on. But $\mathbf{C}_p$ is not locally compact so there is trouble doing that. There is also Coleman integration, which is much more technical and relies on "Frobenius lifting"---see a paper of Colemanfromthe80s called sthg like "p-adic integration". –  Kevin Buzzard Mar 7 '11 at 13:55
    
Using the Haar measure we can do integration on p adic numbers. But only of complex or real valued functions and not functions which attain p-adic values itself. Right? But this I explicitly excluded. –  wood Mar 7 '11 at 14:03
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8 Answers

up vote 22 down vote accepted

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 $$H_a^b=\begin{bmatrix}1& \int_a^bA ;\\ 0& 1\end{bmatrix}$$

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nice answer. do you have any reference of this path dependent line integral of Berkovich. is it published? –  wood Mar 8 '11 at 8:27
    
Yes, apparently, it's Annals of Math Studies 162. –  Minhyong Kim Mar 8 '11 at 13:13
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In recent decade, several number theorists for instance, professor T.Kim, H.Srivastava, Serkan Araci ,extended q-integral concept for p-adic numbers and found several combinatorial identities for Bernoulli, Euler and Genocchi numbers by using this new method. I try to briefly explain this method here ,

Assume that $p$ be a fixed odd prime number. By $Z_p$ we denote the ring of p-adic rational integers, Q denotes the field of rational numbers, $Q_p$ denotes the field of p-adic rational numbers, and $C_p$ denotes the completion of algebraic closure of $Q_p$. Let N be the set of natural numbers and $N^∗ = N ∪ { 0 } $. The p-adic absolute value is defined by $|p|_p =\frac{1}{p}$. We assume $|q − 1|_p < 1 $ as an indeterminate. Let $UD(Z_p)$ be the space of uniformly differentiable functions on $Z_p$. For a positive integer $d$ with $(d, p) = 1$, set

$$X = X_d = lim_{←n} Z/dp^nZ,$$

$$X^∗ = ∪_{0< a< dp , (a,p)=1} a + dpZ_p$$

and $a + dp^nZ_p = \{x ∈ X | x ≡ a (mod dp^n)\}$.

where $a ∈ Z$ satisfies the condition $0 ≤ a < dp^n$.

Firstly, for introducing fermionic p-adic q-integral, we need some basic information which we state here. A measure on $Z_p$ with values in a p-adic Banach space

B is a continuous linear map $$f →\int f(x)\mu =\int_{Z_p} f(x)\mu(x)$$

from $C^0(Z_p,C_p)$, (continuous function on $Z_p$ ) to $B$. We know that the set of locally constant functions from $Z_p$ to $Q_p$ is dense in $C^0(Z_p,C_p)$ so explicitly, for all $f ∈ C^0(Z_p,C_p)$, the locally constant functions $$f_n =\sum_{i=0}^{p^n−1} f(i)1_{i+p^nZ_p} → f$$ in $C^0$.

Now, set $\mu (i + p^nZ_p) = \int_{Z_p}1_{i+p^nZ_p}\mu$, then $\int _{Z_p}f\mu$ is given by the following Riemannian sum,

$$\int_{Z_p}f\mu = lim_{n→∞}\sum_{i=0}^{p^n−1}f(i)\mu(i + p^nZ_p)$$.

T. Kim introduced $\mu$ as follows: $\mu_{−q}(a + p^nZ_p) =\frac{(-q)^a}{[p^n]_{-q}}$ for $f \in UD(Z_p)$, which is famous to the fermionic p-adic q-integral on $Z_p$ and you can find the applications of this definition in several papers which about q-Bernoulli numbers and polynomials .See here

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As Minhyong mentioned, Berkovich has a theory of integration of one-forms on his analytic spaces that yields parallel transport of étale-locally unipotent connections along both paths in the Berkovich-analytic sense (continuous maps from $[0,1]$ to the Berkovich space) and "homotopy classes of étale paths" (which are isomorphisms of the fiber functors at the geometric endpoints). Where Coleman integration depends on a choice of global branch of logarithm, Berkovich packages all possible choices into a sheafy construction.

There is a book with more details.

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One could also approach this question by looking at the theory of distributions such as those that arise in Iwasawa theory. Chapter 13 of Washington's book Introduction to Cyclotomic Fields gives an introduction to this theory. Schneider's Non-Archimedean Functional Analysis gives results on the linear duals of spaces of p-adic functions of various types and these spaces of distributions amount to an integration theory. You could also look at my paper with Schneider called "p-adic Fourier Theory" (and the references therein) for another type of p-adic integration.

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The answer to the short version of your question is: yes, there is a $p$-adic theory of integration.

As to whether an analogue of Cauchy's theorem holds in such a theory, I assume you are thinking of Cauchy's integral formula, and I work with the theory of Coleman. That formula amounts to the statement that the complex logarithm is a multivalued function. But over the $p$-adics, for each choice of $\mathscr{L}\in\mathbb{C}_p$ there is a so-called branch of the $p$-adic logarithm, which is defined on all $\mathbb{C}_p^\times$, uniquely determined by the fact that it behaves as expected for the product and it extends the convergent power series

$$\log(1+x)=\sum_{n=1}^\infty\frac{(-x)^n}{n}$$

of $x\in\mathbb{C}_p$ with $v_p(x)>0$. Each of these provides a primitive of the function $f(x)=\frac{1}{x}$ on $\mathbb{C}_p$, and the ``Fundamental Theorem of Calculus´´ satisfied by Coleman's theory of $p$-adic line integrals, will tell you that the integral of $f(x)$ along a closed loop around $0\in\mathbb{C}_p$ is zero.

A good reference:

MR0782557 (86j:14014) Coleman, Robert F. Torsion points on curves and $p$-adic abelian integrals. Ann. of Math. (2) 121 (1985), no. 1, 111–168.

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Christophe Breuil's Bourbaki report Intégration sur les variétés p-adiques (Séminaire Bourbaki 860, Astérisque 266, 2000, 319--350) might be a good place to start looking for answers. It is available on his webpage and also at Numdam (Grenoble).

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There exists a theory of the Shnirelman integral providing Cauchy-type formulas for $\mathbb C_p$-valued rigid (Krasner) analytic functions on subsets of $\mathbb C_p$. For a modern exposition see M. M. Vishik, Non-Archimedean spectral theory, J. Soviet Math. 30 (1985), 2513--2554.

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See also N. Koblitz, p-Adic Analysis. A Short Course on Recent Work. Cambridge University Press, 1980. –  Anatoly Kochubei Mar 7 '11 at 15:02
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I recall that I once looked at a book by Schikhof called Ultrametric calculus. An introduction to $p$-adic analysis, which is available from CUP and might be what you're looking for.

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