What is a p-adic metric?

I love learning about new ways to do geometry. Or rather, about strange new ways to redefine what geometry might mean. Peter Scholze published lectures about p-adic geometry, and this series of posts are the result of my feeble attempts to get a bit closer to understanding what these are about! In fact, Scholze is merely responsible for some recent progress in the area, the idea is well over a century old. I hope to cover some of his work in future posts, though I have to admit that that seems far away the moment.

Other relevant names are Lafforgue, ... TO BE WRITTEN

Ever since Riemann, we know that geometry is defined via a metric. So what's the metric in p-adic geometry?

P-adic what?

Kurt Hensel first described p-adic numbers in 1897, although it seems fair to say that Ernst Kummer used them earlier.

https://en.wikipedia.org/wiki/P-adic_number

The p-adic number systems are an alternative way to extend the idea of fractions, different from the real numbers. Yes, are, because there are many. One for each prime. This is one way in which they are different from the real numbers: they are the same no matter in which base you look at them, and in which non-prime bases are perfectly okay.

Another difference is that p-adic numbers are typically written with their possibly infinitely many digits to the left. As consolation we won't need a decimal point for them. Nor a minus sign!

The p-adic numbers are popular among number theorists, because they allow generalizing the methods of calculus to the integers, basically extending the integers, adding new ones which have infinitely many digits. Indeed, they were originally motivated by an attempt to make power series work for number theory.

https://en.wikipedia.org/wiki/P-adic_analysis

They form a commutative ring. To describe how that works, it's helpful to introduce a property: the p-adic valuation $v(x)$. It is the number of zeros at the right end of a p-adic number. Using $v(x)$ we can conveniently define the invertible p-adics (or units) as those with valuation zero. This gives an integral domain (which is almost a division ring) with many nice properties!

Enough introduction, show me the metric!

$$d_p(m) = \frac{1}{p^n}$$

where p is the base of our p-adic system, and n is the number of times one can divide m by p. We can use the valuation function to avoid explaining the meaning of n, because dividing by $p$ amounts to shifting the digits one place to the right:

$$d_p(m) = \frac{1}{p^{v(m)}}$$

Further, the distance between two numbers is still something of a difference:

$$d_p(a-b)$$

Let's first try this out with simple integers, which don't extend infinitely far to the left as p-adic numbers might:

$$d_2(4-3) = d(1) = \frac{1}{2^0} = 1 \\ d_2(8-4) = d(4) = \frac{1}{2^2} = \frac{1}{4} \\$$

So 8 and 4 are closer to each other than 4 and 3!

Let's have a look at a nice picture of p-adic numbers:

[[[ symmetric image of a ternary tree ]]] TO BE WRITTEN

Note that subtracting similar numbers, which have some digits on the right which are the same, yields a number which ends in that many zeros. The more zeros a number has on the right end, the smaller it is!

Wait, what?

Just like commonplace integers, a p-adic number, which keeps having digits to the left, can be written as a sum of powers of p.

$$...21200 \\ = ... 2*3^4 + 1*3^3 + 2*3^2 + 0*3^1 + 0*3^0 \\ = ... 2*3^4 + 1*3^3 + 2*3^2 \\ = ( ... 2*3^2 + 1*3^1 + 2)*3^2 \\$$

In that sense, a number ending with many zeros is a small number!

This is very similar to the way the smallness of a real number can be seen by counting the number of zeros between the decimal point and the first nonzero digit. We don't usually write the infinite number of zeros to the right of a real number.

Ostrowki's Theorem

Alexander Ostrowki proved in 1916 that there are only these two non-trivial metrics over the rationals, the familiar absolute value, and the p-adic metric. (The only trivial metric gives 0 or 1, depending on wether $x=y$.)

https://en.wikipedia.org/wiki/Ostrowski%27s_theorem

That's surprising, because the requirements for a metric aren't very rigid:

$$1. D(x,y) = 0 \implies x = y \\ 2. D(x,y) = D(y,x) \textrm{ (symmetry)} \\ 3. D(x,y) \leq D(x,z) + D(z,y) \textrm{ (triangle inequality)} \\$$

Turns out, p-adic space is even ultrametric! An ultrametric strengthens the triangle equality to:

$$D(x,z) \leq max(D(x,y),D(y,z))$$

which has some funny consequences: every triange is isosceles! And also that every point inside a ball is its center!

https://en.wikipedia.org/wiki/Ultrametric

The p-adic circle

Now let's have some fun, and construct the p-adic version of a circle! Let's do a one-dimensional circle in $P_2$ at the origin first, say, of radius $1/8$. It consists of all points $z$ for which $d_2(z)=1/8. That is all those points, which end in exactly three zeros. That is a lot of points!

Points which have only two zeros, have radius 1/4, belong to larger circles, and points ending on more than three zeros belong to smaller circles.

Here something funny happens: remember that in an ultrametric space all points inside a circle are all centers? Here's what that means: to measure the distrance of a point to the circle's point, we first subtract them from each other, and then compute the metric. For example:

$$a = ...10000 \textrm{ has metric } 1/16 \\ b = ...11000 \textrm{ has metric } 1/8 \\ a-b = ...01000 \textrm{ has metric } 1/8 \\$$

If $a$ is a point inside the circle with radius $1/8$, then its distance from any point $b$ on the boundary is $1/8$. The same is true for the origin, which is by definition $1/8$ away from the boundary!

For more, watch

Youtube user SuperScript published a nice intro into p-adic numbers, 'How to wrangle infinity'

[youtube]

However, my post is an approach to p-adic geometry, so his video 'intuition for the p-adic metric' is where things get interesting:

[youtube]

And here's a second video with more on distances, 'Ostrowki's Theorem (p-adic metric continued)':

[youtube]

for more, read

Bernard Le Stum, 'One century of p-adic geometry – From Hensel to Berkovich and beyond, talk notes, June 2012'

http://www-irma.u-strasbg.fr/IMG/pdf/NotesCoursLeStum.pdf

Peter Scholze, Jared Weinstein, 'Berkeley lectures on p-adic geometry'

http://www.math.uni-bonn.de/people/scholze/Berkeley.pdf

The n-Lab on 'non-archimedean analytic geometry', which has a nice introduction to non-archimedean geometry:

https://ncatlab.org/nlab/show/non-archimedean+analytic+geometry

An archimedean group is a linearly ordered group in which every positive element is bounded above by a natural number.

One can even do homotopy or homology in p-adic geometry:

• https://ncatlab.org/nlab/show/p-adic+homotopy
• https://ncatlab.org/nlab/show/p-adic+cohomology