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
The goal of this post is to introduce, in a very informal way, the notion of a reductive group, and discuss some examples.
Algebraic Geometry, Algebraic Groups, Shimura varieties, Uncategorized and tagged Algebraic Geometry, algebraic groups, Intuition, linear algebraic groups, overview, Reductive groups, semisimple groups on .
February 27, 2016 5 Comments
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.
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.
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.
Class field theory, Number Theory and tagged class field theory, Intuition, local class field theory, lubin-tate formal group law, lubin-tate theory, motivation, number theory, proof, proof overview, relation to Langlands on .
September 1, 2015
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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.
Algebraic Geometry, Etale Cohomology, Number Theory and tagged abelian varieties, arithmetic geometry, etale cohomology, kummer theory, mordell-weil theorem, mordell-weil theorem proof, principal homogenous space, proof, torsor, Weak Mordell-Weil theorem on .
August 17, 2015 2 Comments
In this post we discuss the basic theory of p-divisible groups, their relationship to formal groups, and the Serre-Tate theorem.
Algebraic Geometry, Algebraic Groups, Galois Representations, Number Theory and tagged abelian varieties, Algebraic Geometry, algebraic groups, arithmetic geometry, crystalline cohomology, finite flat group schemes, formal groups, Formal schemes, Formal schemes intuition, number theory, p-adic Hodge theory, p-divisible groups, Serre-Tate theorem on .
August 10, 2015 2 Comments