I really wanted to move on from this topic, because the theory gets much more interesting when we move to -divisible groups over some larger rings than just algebraically closed fields. Unfortunately, while looking over how Demazure builds the theory in Lectures on -divisible Groups, I realized that it would be a crime to bring you this far and not concretely show you the power of thinking in terms of Newton polygons.
As usual, let’s fix an algebraically closed field of positive characteristic to work over. I was vague last time about the anti-equivalence of categories between -divisible groups and -crystals mostly because I was just going off of memory. When I looked it up, I found out I was slightly wrong. Let’s compute some examples of some slopes.
Recall that and . In particular, , so in our -crystal theory we get that the normalized -adic valuation of the eigenvalue of is . Recall that we called this the slope (it will become clear why in a moment).
Our other main example was with . In this case we have is “the” eigenvalue which has -adic valuation . These slopes totally determine the -crystal up to isomorphism, and the category of -crystals (with slopes in the range to ) is anti-equivalent to the category of -divisible groups.
The Dieudonné-Manin decomposition says that we can always decompose as a direct sum of vector spaces indexed by these slopes. For example, if I had a height three -divisible group, would be three dimensional. If it decomposed as where was -dimensional (there is a repeated -eigenvalue of slope ), then would be -dimensional, and I could just read off that my -divisible group must be isogenous to .
In general, since we have a decomposition where is the part with slopes strictly in we get a decomposition where is the dimension of and does not have any factors of those forms.
This is where the Newton polygon comes in. We can visually arrange this information as follows. Put the slopes of in increasing order . Make a polygon in the first quadrant by plotting the points , , … , .
This might look confusing, but all it says is to get from to make a line segment of slope and make the segment go to the right for . This way you visually encode the slope with the actual slope of the segment, and the longer the segment is the bigger the multiplicity of that eigenvalue.
But this way of encoding the information gives us something even better, because it turns out that all these must have integer coordinates (a highly non-obvious fact proved in the book by Demazure listed above). This greatly restricts our possibilities for Dieudonné -crystals. Consider the height case. We have is two dimensional, so we have slopes (possibly the same). The maximal coordinate you could ever reach is if both slopes were maximal which is . In that case you just get the line segment from to . The lowest you could get is if the slopes were both in which case you get a line segment to .
Every other possibility must be a polygon between these two with integer breaking points and increasing order of slopes. Draw it (or if you want to cheat look below). You will see that there are obviously only two other possibilities. The one that goes to to which is a slope and slope and corresponds to and the one that goes to . This corresponds to a slope with multiplicity . This corresponds to the for supersingular elliptic curves. That recovers our list from last time.
We now just have a bit of a game to determine all height -divisible groups up to isogeny (and it turns out in this small height case that determines them up to isomorphism). You can just draw all the possibilities for Newton polygons as in the height case to see that the only possibilities are , , , , and then two others: which corresponds to the thing with a triple eigenvalue of slope and which corresponds to the thing with a triple eigenvalue of slope .
To finish this post (and hopefully topic!) let’s bring this back to elliptic curves one more time. It turns out that . Without reminding you of the technical mumbo-jumbo of crystalline cohomology, let’s think why this might be reasonable. We know is always height , so is rank . But if we consider that crystalline cohomology should be some sort of -adic cohomology theory that “remembers topological information” (whatever that means), then we would guess that some topological of a “torus” should be rank as well.
Moreover, the crystalline cohomology comes with a natural Frobenius action. But if we believe there is some sort of Weil conjecture magic that also applies to crystalline cohomology (I mean, it is a Weil cohomology theory), then we would have to believe that the product of the eigenvalues of this Frobenius equals . Recall in the “classical case” that the characteristic polynomial has the form . So there are actually only two possibilities in this case, both slope or one of slope and the other of slope . As we’ve noted, these are the two that occur.
In fact, this is a more general phenomenon. When thinking about -divisible groups arising from algebraic varieties, because of these Weil conjecture type considerations, the Newton polygons must actually fit into much narrower regions and sometimes this totally forces the whole thing. For example, the enlarged formal Brauer group of an ordinary K3 surface has height , but the whole Newton polygon is fully determined by having to fit into a certain region and knowing its connected component.