I would normally take $p$-adic integration to mean "integration of $p$-adic valued functions" or "integration of differential forms with some kind of $p$-adic valued functions as coefficients", where the integration is also taking place over some kind of $p$-adic space or manifold.

The reason for wanting such theories are various. One reason is indicated in George S.'s answer: there are *known* analogues of classical Hodge theory, known as $p$-adic Hodge theory, whose proofs however are not analytic, but rather proceed via arithmetic geometry. One would like to have more analytic ways of thinking about them, and this is one goal of Robert Coleman's theory. (In a recent volume of Asterisque, namely vol. 331, Coleman and Iovita have an article, "Hidden structures on semistable curves", related to this problem.) (Note also that $p$-adic Hodge theory relates $p$-adic etale cohomology to crystalline cohomology, which gives on answer to your question of how $p$-adic integration might be related to those topics.)

Another reason is that many integral formulas (involving usual archimedean integrals) appear in the theory of classical $L$-functions attached
to automorphic forms, and one would like, at least in certain contexts, to be able to write down $p$-adic analogues so as to construct $p$-adic $L$-functions.

As for what machinery is used: in the theory of $p$-adic $L$-functions and related contexts in Iwasawa theory, often nothing more is used than basic computations with Riemann sums. In the material related to $p$-adic Hodge theory, much more substantial theoretical foundations are used: tools from arithemtic geometry, rigid analysis, possibly Berkovich spaces, and related topics.

## Best Answer

In the context of Colmez's papers, the notation has its own meaning, not related (by more than vague analogy) to other meanings it has in other contexts where it is used.

You will have to read Colmez's article in Asterisque 330 to learn the details.

Roughly: you should think of the $(\varphi,\Gamma)$-module as being an object (like a space of measures, or functions) living over $\mathbb Z_p$. Then $D\boxtimes \mathbb Q_p$ is what you get by using scaling by $p$ (which is rigorously defined using the operator $\psi$) to "stretch" the $(\varphi,\Gamma)$-module out over $\mathbb Q_p$.

Similarly $D\boxtimes \mathbb P^1$ is what you by taking two copies of $D$ and gluing them together, in accordance with the way that $\mathbb P^1(\mathbb Q_p)$ is obtained by gluing together two copies of $\mathbb Z_p$.

Non-mathematical remark:I should add that what you are asking about is very recent mathematics, and has a pretty high entry-level. Where/with who are you learning this material? You may be better off asking your advisor directly rather than trying to learn this on math.SE.You may also want to look at some of Colmez's lectures, several of which should be available online. He lectured this past July at the Durham conference, and I believe those lectures were videotaped. In the past he has lectured at Luminy (several times, I think), at the Newton Institute (Summer of 09, if I remember correctly), and this past March he gave a lecture course at the IAS (although I wasn't there, so I don't know if it was filmed).

You may also find it easier to study the functor from $GL_2$-reps. to Galois reps. before trying to go backwards from Galois reps. to $GL_2$-reps. (which is the point of the $\boxtimes$ constructions). As well as Colmez's Asterisque 330 article, there is also my short preprint

On a class of coherent rings ..., which you will be able to find with a google search.