# Intuitive pictures in characteristic p

This is a tough one, but does anyone know of any images that recall characteristic p geometry (over algebraically closed fields) in some sense? It is not enough if it is some picture that can be also understood solely in characteristic 0.

A quick search through the literature has proved fruitless.

I have been thinking for a while of asking this question, but I never had a pressing need rather than my own curiosity. However, now I am trying to improve a poster by including pictures but the topic is algebraic geometry in characteristic p.

An example of such an image in Complex Geometry would be the arrangement of contracted and blow up curves in the standard Cremona transformation of the projective plane. The one in this poster is simple but effective.

I believe this question also might be of interest for people who try to explain research to non-mathematicians or simply to mathematicians who are not geometers.

• surely this one would qualify: en.wikipedia..org/wiki/Fano_plane Commented Dec 20, 2012 at 12:19
• My bad, I should have stated algebraically closed fields. I did that now. In that case the Fano plane is not applicable. Commented Dec 21, 2012 at 11:35

I don't think you can draw something meaningful - I would be surprised if someone made a good drawing of the Frobenius morphism ;).

That being said, here is an example (possibly misleading or unrelated to your research) I saw in the slides of Benedict Gross's lectures on the arithmetic of hyperelliptic curves. Take a prime $p$, say $p=57$, and an equation of an hyperelliptic curve $y^2 = x^n + ax^{n-2} + \ldots$ with integer coefficients. Draw a $p\times p$ square and mark the solutions to the above equation mod $p$. The resulting picture exhibits the following:

• It is a mixture of chaos and geometry: there is a visible symmetry coming from the hyperelliptic involution $(x, y)\mapsto (x, -y)$.
• The solutions form a finite set, in particular, it makes combinatorial arguments possible. We can ask how many points are there and whether it gives us some "geometric" information. This is not obvious to someone from other fields, or a non-mathematician.

You can include drawings of the same curve over $\mathbb{R}$ and $\mathbb{C}$. I think the equation $y^2 = \ldots$ and the three pictures together explain pretty well what algebraic geometry is about without going into too much detail.

• +1 For the Grothendieck prime ;) Commented Dec 20, 2012 at 11:29
• I certainly can't make a good drawing of anything at all, but my picture of a separable map of, say, algebraic surfaces, is of the sea with a wave breaking along the ramification divisor, while a purely inseparable map is the sea where waves are breaking everywhere at once. Honestly, I find this at least slightly illuminating. Commented Dec 20, 2012 at 13:46
• For me, an inseparable map (of curves, say), looks like an ordinary map of curves except that, whenever I look at a particular point closely, I see that the map is rapidly contracting (like $x \mapsto x^3$ on the reals near $0$). I don't see any "folding over" or "wave breaking" because the map is still bijective. Unfortunately, it's hard to draw an image which changes based on where my eyeball is focused. Commented Dec 20, 2012 at 17:08
• Out of curiousity: Let $X$ be the interval $[0,1]$ with the standard metric and let $Y$ be the interval $[0,1]$ with the metric $d(x,y) = (x-y)^2$. How do you visualize the map of metric spaces which is the identity on points? Because that has the same sort of issues as a inseparable map in characteristic $p$. Commented Dec 20, 2012 at 17:09
• @David, $d(x,y)=(x-y)^2$ doesn't induce a metric on $[0,1]$. Commented Dec 20, 2012 at 20:11

I heard the following analogy when talking to some specialists in absolute de Rham theory. I think Deninger's name was mentioned at about the same time.

One possible way to imagine a variety over $\mathbb{F}_p$ is as a manifold equipped with a distinguished vector field, which we call "Frobenius". The usual discrete Frobenius that admits integer powers is the unit time evolution of the flow. Orbits of points on the variety that are defined over finite fields correspond to closed integral curves of the flow, and we assume that such curves only have integral periodicity.

There are a few problems. If there are points defined over an imperfect field, you may have to consider flows that start somewhere, like a distinguished submanifold. Also, it is not clear to me how one connects such a picture to the set of solutions of a system of polynomial equations.

Edit: (response to Daniel Litt's comment) I must confess that I am not particularly familiar with the idea, so I can only fill in vague guesses. Also, I don't know why one would want to interpolate between different powers of Frobenius. The fundamental idea seems to be that if we examine the spectrum of a finite field using étale glasses rather than Zariski or Nisnevich glasses, it looks a lot like a circle, since the étale fundamental group is a completion of $\mathbb{Z}$. This suggests that if we were to propose some real geometric object as an analogue of a variety, $\mathbb{F}_q$-points should be distinguished circles, perhaps with some distinguished automorphism.

The picture of finite fields as circles also shows up in "arithmetic topology" speculation for similar reasons. Here, the spectra of number rings are viewed as 3-manifolds, and the finite points are distinguished embedded circles, for which some kind of linking number may be defined homologically. As far as I know, this is another analogy that seems to be waiting for a substantial application.

• Could you expand a bit on this idea? E.g. is there some way to interpolate between different powers of Frobenius? If not this seems like a strange analogy... Commented Dec 21, 2012 at 19:08
• This sounds pretty amazing! I wish there was an example where keeping this picture in mind allows you to predict a couple of true facts.
– LMN
Commented Dec 21, 2012 at 19:41
• When I heard Deninger talk about this stuff, he made the following point, which I found very compelling. The idea that a number field is like a 3-manifold, and the primes within it like embedded 1-manifolds, is very old, and has lots of appealing features. The question, then is -- in real life, WHY would a 3-manifold M have a canonical countable set of embedded 1-manifolds? And the only reason one can think of is that M must be endowed with a flow!
– JSE
Commented Dec 22, 2012 at 4:36
• That is a good point, JSE. I may have mixed up causes with effects. Commented Dec 22, 2012 at 7:16
• Thanks for this answer, but do I understand that you are assuming the field to be finite, or does it work for algebraically closed fields too? Commented Dec 22, 2012 at 13:11

This is not nearly as nice an example as the others, but I always imagined the line in characteristic five geometry as a countable set of points that glow like blue Christmas tree lights vaguely in the shape of a narrow paraboloid, with 0 at the vertex, with 1, 2,3, and 4 at the next "height", with the quadratic "irrationals" next, etc. although the makes it seem like each field of order $p^n$ contains the field of order $p^{n-1}$. Like in Carnahan's answer, I imagine the Frobenius automorphism shuffling around everything in each fixed ring. Finally, I imagine the Zariski topology as a glowy light filling in the paraboloid representing the forces that each point exerts on all others; it's constant everywhere, as in the Zariski topology, there's no real sense of distance.

More complicated varieties I imagine as a double cone with glowy lights (two Christmas trees!) or as the paraboloid distortedt to have self-intersections in some of the lights. something like the galactic gravity simulation or spherical pendulum in this website

• This is a very seasonally appropriate picture. Commented Dec 22, 2012 at 4:46
• I particularly like this interpretation. Did you come with it by yourself or is it well known? Commented Dec 22, 2012 at 13:13
• I came to it myself, but I think it's based on the Red Book of Varieties' picture of $\mathbb{Z}$[x]. Commented Dec 22, 2012 at 14:36