# What is the operation $\boxtimes$?

Reading papers about $p$-adic analysis and Galois representations, I have found objects like this $D \boxtimes \mathbb{Q}_p$. So my question is what is $\boxtimes$ and how do we read it ?

• Can you give some more context? What is $D$? – Qiaochu Yuan Jul 7 '11 at 16:19
• $D$ is a $(\phi,\Gamma)$-module. This notation is often used in Pierre Colmez's papers. He talks also about $D \boxtimes U$ where $U$ is an open subset of $Q_p$. – user10676 Jul 7 '11 at 16:36
• Maybe you should ask Pierre Colmez, then. – Gerry Myerson Jul 8 '11 at 1:51
• At mathoverflow.net/questions/69222/h4-of-the-monster it's used for "the fusion of twisted reps," if that's any help. – Gerry Myerson Jul 8 '11 at 1:55

## 3 Answers

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.

There is a notion of external (also called exterior, or box) tensor product $\boxtimes$ ( e.g. http://books.google.com/books?id=6GUH8ARxhp8C&pg=PA24 ).

I think that the usage is not completely standardized, in that the definition is often adapted to other contexts (examples at https://mathoverflow.net/search?q=boxtimes ), but the adaptations are not always consistent with each other.

• Dear zyx, Yes, this is rather standard notation, but the way Colmez uses $\boxtimes$, it is not really an exterior tensor product. The point is that the $(\varphi,\Gamma)$-module $D$ is a vector space --- which can be thought of as a space of sections of a sheaf over $\mathbb Z_p$ --- while the $\mathbb Q_p$ and $\mathbb P^1$ that appear in the expression $D\boxtimes ?$ are spaces extending the original domain $\mathbb Z_p$ of the sheaf. So in Colmez's use of the notation, it is not at all an example of two objects of the same kind being tensored together. Regards, – Matt E Sep 6 '11 at 18:11
• @Matt: I should have made it clearer -- your answer was definitive as far as the OP question on Colmez' paper is concerned, and I am just offering the additional cultural observation that abuses (or nonstandard uses) of this notation are actually fairly common and sometimes more confusing than writing a box product with P^1 (when sheaves canonically extend there etc). My recollection of other papers by Colmez is that he is good about providing definitions, but other authors use their own versions of the $\boxtimes$ notation without comment, leaving an interesting puzzle for the reader. – zyx Sep 6 '11 at 18:58

This is probably not directly what you're asking about, but it might be related. In any event it can't hurt to add it.

Let $X$ be a topological space and let $E_1 \to X$, $E_2 \to X$ be vector bundles (or sheaves, probably). One often defines $S_1 \boxtimes S_2$ to be the bundle $\pi_1^\ast E_1 \otimes \pi_2^\ast E_2^\ast$ over $X \times X$, where $\pi_1, \pi_2: X \times X \to X$ are the usual projection maps. Thus a vector over the point $(p,q)$ is a linear map from $S_1(p)$ to $S_1(q)$. This bundle is useful in differential geometry because its sections are Schwarz kernels of linear operators. I have seen it arise in algebraic geometry (over $\mathbb{C}$) as well for a similar reason. Perhaps there is an analogy between the notation coming from geometry and the notation coming from representation theory and number theory?

• Dear Paul, As I wrote in my comment to zyx's answer, in this particular context I the objects being "multiplied" via $\boxtimes$ are not two vector bundles (or any two objects of the same type), but rather (a space of global sections of) an equivariant sheaf and a certain topological space extending the domain of this sheaf. So it is not particularly analogous to an exterior product of vector bundles. Regards, – Matt E Sep 7 '11 at 2:26