This is a rough transcription of a talk I gave to a class of algebraic number theory students at UC Berkeley with the goal of trying to understand how one might bring to bear modern techniques in number theory/geometry on some classical questions. I have essentially kept the format the same, while adding a bit of extra material (and adding in their responses to questions I asked).

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# Etale Cohomology

# Some motivation for p-adic Hodge theory

These are some notes that I wrote for a learning course at Berkeley–the goal being to understand the statement of the global Langlands conjecture.

The goal of the talk (that these notes were written for) was, specifically, to motivate -adic Hodge theory with an eye, in particular, towards where it might be useful in understanding the statement of Langlands.

These are even less edited than usual, so I profusely apologize for any mistakes. As always, corrections/comments are very welcome!

# A computation a day: a pullback pushforward

In this post we compute the Galois representation where is the natural inclusion and is the inclusion of the origin.

# Around abelian schemes over the integers

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 .

# 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.

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

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

# A computation a day: compactly supported cohomology

In this post we compute the compactly supported cohomology of some simple varieties.