This is the transcription to blog format of a talk I gave at the UC Berkeley Student Arithmetic Geometry Seminar about several topics related to Fontaine’s famous result that there are no abelian schemes over .

# Reductive groups: a rapid introduction

The goal of this post is to introduce, in a very informal way, the notion of a reductive group, and discuss some examples.

# Maps from simply connected projective varieties to curves

In this post we prove a general result that shows, in particular, that any map from a simply connected to a curve of genus at least is constant.

# Shimura Varieties: motivation

This will be the first in a series of posts discussing Shimura varieties. In particular, we will focus here on a sort of broad motivation for the subject—why Shimura varieties are a natural thing to study and, in particular, what they give us.

# Local class field theory: a discussion

In this post we discuss local class field theory (specifically looking at -adic fields) with a focus on the broader picture, and the multiple approaches.

# Kummer theory and the weak Mordell-Weil theorem

In this post we discuss the notion of Kummer theory in its general form, and how this leads to a proof of the (weak) Mordell-Weil theorem.

# p-divisible groups, formal groups, and the Serre-Tate theorem

In this post we discuss the basic theory of p-divisible groups, their relationship to formal groups, and the Serre-Tate theorem.