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.