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.

(more…)

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.

(more…)

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.

(more…)

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).

(more…)