I came across this idea a long time ago, but I needed the result that uses it in its proof again, so I was curious about figuring out what in the world is going on. It turns out that you can make “-adic measures” to integrate against on algebraic varieties. This is a pretty cool idea that I never would have guessed possible. I mean, maybe complex varieties or something, but over -adic fields?

Let’s start with a pretty standard setup in -adic geometry. Let be a finite extension and the ring of integers of . Let be the residue field. If this scares you, then just take and .

Now let be a smooth scheme of relative dimension . The picture to have in mind here is some smooth -dimensional variety over a finite field as the closed fiber and a smooth characteristic version of this variety, , as the generic fiber. This scheme is just interpolating between the two.

Now suppose we have an -form . We want to say what it means to integrate against this form. Let be the normalized -adic valuation on . We want to consider the -adic topology on the set of -valued points . This can be a little weird if you haven’t done it before. It is a totally disconnected, compact space.

The idea for the definition is the exact naive way of converting the definition from a manifold to this setting. Consider some point . Locally in the -adic topology we can find a “disk” containing . This means there is some open about together with a -adic analytic isomorphism to some open.

In the usual way, we now have a choice of local coordinates . This means we can write where is a -adic analytic on . Now we just define

Now maybe it looks like we’ve converted this to another weird -adic integration problem that we don’t know how to do, but we the right hand side makes sense because is a compact topological group so we integrate with respect to the normalized Haar measure. Now we’re done, because modulo standard arguments that everything patches together we can define in terms of these local patches (the reason for being able to patch without bump functions will be clear in a moment, but roughly on overlaps the form will differ by a unit with valuation ).

This allows us to define a “volume form” for smooth -adic schemes. We will call an -form a volume form if it is nowhere vanishing (i.e. it trivializes ). You might be scared that the volume you get by integrating isn’t well-defined. After all, on a real manifold you can just scale a non-vanishing -form to get another one, but the integral will be scaled by that constant.

We’re in luck here, because if and are both volume forms, then there is some non-vanishing function such that . Since is never , it is invertible, and hence is a unit. This means , so since we can only get other volume forms by scaling by a function with -adic valuation everywhere the volume is a well-defined notion under this definition! (A priori, there could be a bunch of “different” forms, though).

It turns out to actually be a really useful notion as well. If we want to compute the volume of , then there is a natural way to do it with our set-up. Consider the reduction mod map . The fiber over any point is a -adic open set, and they partition into a disjoint union of mutually isomorphic sets (recall the reduction map is surjective here by the relevant variant on Hensel’s lemma). Fix one point , and define . Then by the above analysis we get

All we have to do is compute this integral over one open now. By our smoothness hypothesis, we can find a regular system of parameters . This is a legitimate choice of coordinates because they define a -adic analytic isomorphism with .

Now we use the same silly trick as before. Suppose , then since is a volume form, can’t vanish and hence on . Thus

This tells us that no matter what is, if there is a volume form (which often there isn’t), then the volume

just suitably multiplies the number of -rational points there are by a factor dependent on the size of the residue field and the dimension of . Next time we’ll talk about the one place I know of that this has been a really useful idea.