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

(more…)# 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

**EDIT:** While these notes might be still useful to read, if one wants a more in-depth explanation of the ideas below see the notes from this post.

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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.

# Another Basic 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. This perspective is certainly implicitly contained within all major texts on the topic, but is less emphasized as a good source of intuition.

# Algebraic de Rham cohomology and the Degeneration of the Hodge spectral sequence

In this post we will discuss various properties of the algebraic de Rham cohomology of a variety . We will focus, in particular, on various aspects of when the Hodge-to-de Rham spectral sequence on the first page, the most interesting case of which happens in positive characteristic.

# A computation a day: the Brauer group of a number ring

In this post we compute the group where is a number field.

# Morphisms of a ‘set theoretic nature’

In this post we characterize morphisms which are determined by their ‘set theoretic’ underpinnings.