algebraic geometry

## The Functor of Points Revisited

Mike Hopkins is giving the Milliman Lectures this week at the University of Washington and the first talk involved this idea that I'm extremely familiar with, but am also surprised at how unfamiliar most mathematicians are with it. I've made almost this exact post several other times, but it bears repeating. As I basked in… Continue reading The Functor of Points Revisited

## An Application of p-adic Volume to Minimal Models

Today I'll sketch a proof of Ito that birational smooth minimal models have all of their Hodge numbers exactly the same. It uses the $latex {p}&fg=000000$-adic integration from last time plus one piece of heavy machinery. First, the piece of heavy machinery: If $latex {X, Y}&fg=000000$ are finite type schemes over the ring of integers… Continue reading An Application of p-adic Volume to Minimal Models

I came across this idea a long time ago, but I needed the result that uses it in its proof again, so I was curious about figuring out what in the world is going on. It turns out that you can make "$latex {p}&fg=000000$-adic measures" to integrate against on algebraic varieties. This is a pretty… Continue reading Volumes of p-adic Schemes

## BSD for a Large Class of Elliptic Curves

I'm giving up on the p-divisible group posts for awhile. I would have to be too technical and tedious to write anything interesting about enlarging the base. It is pretty fascinating stuff, but not blog material at the moment. I've been playing around with counting fibration structures on K3 surfaces, and I just noticed something… Continue reading BSD for a Large Class of Elliptic Curves

## Newton Polygons of p-Divisible Groups

I really wanted to move on from this topic, because the theory gets much more interesting when we move to $latex {p}&fg=000000$-divisible groups over some larger rings than just algebraically closed fields. Unfortunately, while looking over how Demazure builds the theory in Lectures on $latex {p}&fg=000000$-divisible Groups, I realized that it would be a crime… Continue reading Newton Polygons of p-Divisible Groups

## More Classification of p-Divisible Groups

Today we'll look a little more closely at $latex {A[p^\infty]}&fg=000000$ for abelian varieties and finish up a different sort of classification that I've found more useful than the one presented earlier as triples $latex {(M,F,V)}&fg=000000$. For safety we'll assume $latex {k}&fg=000000$ is algebraically closed of characteristic $latex {p>0}&fg=000000$ for the remainder of this post. First,… Continue reading More Classification of p-Divisible Groups

## A Quick User’s Guide to Dieudonné Modules of p-Divisible Groups

Last time we saw that if we consider a $latex {p}&fg=000000$-divisible group $latex {G}&fg=000000$ over a perfect field of characteristic $latex {p>0}&fg=000000$, that there wasn't a whole lot of information that went into determining it up to isomorphism. Today we'll make this precise. It turns out that up to isomorphism we can translate $latex {G}&fg=000000$… Continue reading A Quick User’s Guide to Dieudonné Modules of p-Divisible Groups