p-adics manifolds with boundary?

Stream: maths

[Floris van Doorn]

This came up during the discussion in #18403. Is there some notion of p-adic manifold with boundary that exists in mathematics (that we want to support in the definition of SmoothManifoldWithCorners)?

[Johan Commelin]

I'm not aware of it. But cc @Kevin Buzzard and @Antoine Chambert-Loir

[Kevin Buzzard]

I think that p-adic manifolds only have very limited usage in mathematics (I think Serre developed some very basic theory) and I don't think there's any useful notion of a p-adic manifold with boundary as there is no <= on the p-adic numbers. The way p-adic geometry is done is not via the manifolds set-up, so I wouldn't bother worrying too much about these things. For example for FLT I will not need p-adic manifolds at all, as far as I know, I'll need some related things but they will need to be defined from scratch rather than using what we already have. The way the theory is treated in the p-adic case is via the theory of sheaves, where the differentiability/analyticity conditions are in the spaces of rings rather than on the underlying topological space.

[Antoine Chambert-Loir]

I agree with Kevin about the case of p-adic manifolds with boundary, but not with the idea that p-adic manifolds have limited usage. They're useful — at least in the theory of p-adic Lie groups. What Serre's (beautiful) theorem indicates, though, is that they have no serious global significance.

[Antoine Chambert-Loir]

Theorem (Serre). For every nonempty compact $p$-adic manifold $V$ of (pure) dimension $d$, there exists a unique integer $r\in\{1,\dots,p-1\}$ such that $V$ is isomorphic to the disjoint union of $r$ $d$-dimensional unit balls $(\mathbf Z_p)^d$.

[Antoine Chambert-Loir]

The proof of existence is not so difficult: first of all, you can use the nonarchimedean condition to write $V$ as a disjoint union of balls; then whenever you have $p$ balls, use the fact that $\mathbf Z_p$ is the disjoint union of $p$ translates $a + p \mathbf Z_p$ to reduce to the case where the number of balls is at most $p-1$. It is nonzero, because $V$ is nonempty. That proves existence.

[Antoine Chambert-Loir]

Uniqueness is slightly more tricky. Because of compactness, $V$ carries Lebesgue-type measures, that is measures that look like, locally, $f(x_1,\dots,x_n) dx_1\cdots dx_n$ (the dimension is now $n$), where $f$ is a nonvanishing continuous function valued in $\mathbf Q_p$. Take such a measure and compute the volume of $V$. This is a $p$-adic number which more or less gives you the number of balls once you have reduced modulo $p-1$.

[Jz Pan]

What is the boundary of, for example, $\mathbb{Z}_p$? Is it $\mathbb{Z}_p^\times$? Then it's isomorphic to p - 1 copies of $\mathbb{Z}_p$...

[Kevin Buzzard]

I think the conclusion of the above is that there's no such thing as the boundary of a p-adic manifold

[Antoine Chambert-Loir]

I concur with Kevin. Another thing to observe is that the $p$-adic unit disk can be defined as $|x|\leq 1$, and then its boundary looks as if it is $|x|=1$. But it can also be defined as $|x-1|\leq 1$, which makes its boundary look as if it is $|x-1|=1$. Both description can't be simultaneously true. If you require, moreover, that the boundary is invariant under analytic diffeomorphisms (defined by a power series, as well as its inverse), and that it's not all of it, then it has to be empty. (I just made this up, it must be easy.)