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.

# Number Theory

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

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

# The Tate Conjecture over finite fields

In this post we will describe the Tate conjecture for varieties over finite fields, focusing on its various forms, and talking about some applications.

# A class field theoretic phenomenon

In this post we discuss one example of what’s called a ‘class field theoretic phenomenon’. In particular, we focus on the application of trying to understand the property of when has three distinct roots modulo , for various primes .

# Some examples of Geometric Galois representations

In this post we discuss the Galois representation associated to a projective scheme , where is a number field. We also discuss how this representation can be computed in several simple cases.

# An invitation to p-adic Hodge theory, or: How i learned to stop worrying and love fontaine

This is the rough outline of a talk I recently gave at the Berkeley Student Algebraic Geometry Seminar on the progression of ideas that might lead one to define the Hodge-Tate decomposition.