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.

# 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.

# A Different Viewpoint on Étale Cohomology

In this post, I would just like to discuss a slightly different perspective on the étale cohomology of varieties. This might be called the ‘relative’ or ‘monodromy’ perspective, and it is rife with geometric intuition. While certainly first principles in some regards, it’s a point of view that I humbly believe is not emphasized well in most basic texts on the subject (e.g. Milne’s *Lectures on Étale Cohomology*).