isocrystals

Stream: maths

[Rob Lewis]

@Johan Commelin requested that we announce this a little while back, but we've been occupied with other things, so a bit delayed -- sorry Johan!

@Heather Macbeth , @Frédéric Dupuis , and I recently wrote a paper on the semilinear maps project. We wanted to show an application of semilinear maps that were not linear or conjugate-linear. Heather suggested defining isocrystals as a target that was in scope, given the Witt vector work. With some prodding and help from Johan and a random MathOverflow post, this turned into a proof of the dimension 1 case of the Dieudonné–Manin classification theorem. Essentially, the 1d case reduces to finding solutions to a sequence of recursively defined polynomials: the hard part is defining those polynomials in a way that we can show they have positive degree.

The PR (#12041) represents all I know about isocrystals and more. (Why do it? Because we can!) But Johan thought this was cool and I trust his taste here :smile: At some point we'll write up a blog post about this.

[Riccardo Brasca]

This is very nice!! I will try to review asap

[Kevin Buzzard]

Indeed this is nice. Now we need p-divisible groups!

[Antoine Chambert-Loir]

Easier would be Hodge and Newton polygons, and Katz's theorem…

[Will Sawin]

Surely the easiest next step would be the arbitrary dimension case of Dieudonne-Manin?

[Johan Commelin]

Shouldn't definitely be in reach