schemes

Weil Restriction and Base Change

The natural way to extend scalars on a scheme is base change, and the natural way to restrict scalars is Weil restriction. To better understand these, we can look at how they interact with each other. In particular: what happens if we take the Weil restriction of a scheme, and then base change back to the original field?

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number fields

Almost All Primes in an Extension have No Inertia over the Base

Let $L/K$ be a Galois extension of number fields. By the Chebotarev density theorem the proportion of primes of $K$ that are totally split in $L$ is $\frac{1}{[L:K]}$ . But from the perspective of $L$ , the situation is different: almost all primes of $L$ lie above a totally split prime of $K$ .

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galois representations

Galois Representations are Determined by a Density 1 Set of Primes

The Brauer-Nesbitt theorem states that a representation $\rho:G\to {\rm GL}_nk$ is uniquely determined (up to semi-simplification) by the characteristic polynomials of $\rho(g)$ for all $g\in G$ . Galois representations are often described in this way, but usually only using the characteristic polynomials of Frobenius elements, and usually away from a finite set of primes, or even on a density 1 set of primes. Let’s check that this is still enough to uniquely determine the representation.

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adic spaces

Adic Spaces II: Analytification and Fiber Products

Now that we’re familiar with the definitions and a few examples from Part I, let’s look at basic operations we can do with adic spaces. As usual let $K$ be a complete non-archimedean field.

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adic spaces

Adic Spaces I: Definitions and Basic Properties

Adic spaces provide a framework for analytic geometry over non-archimedean fields. They were initially developed by Huber in the ’90s, and have become popular following the advent of perfectoid spaces. These notes (and Part II) cover the very basics of the theory.

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elliptic curves

Reduction Types of Elliptic Curves

Reduction $\;{\rm mod}\; p$ is a useful skill to have as an arithmetic geometer. Here we examine some elliptic curves whose reductions can be described relatively easily, and at the end some curious behavior of reduction mod $p$ upon extending the base field. Nearly all of this is from the book A First Course in Modular Forms by Diamond & Shurman, and in particular exercises 8.3.6 and 8.4.4.

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number fields

Frobenius in Galois Groups

Another useful application of Galois theory to prime factorization in number fields is the Frobenius element associated to a prime.

Let $L/K$ be a Galois extension of number fields, and suppose $\mathfrak{p}$ is factorized in $L$ as $\mathfrak{p}=\mathfrak{P}_1^e\cdots\mathfrak{P}_r^e$ with each $\mathfrak{P}_i$ having inertia degree $f$ . That is, $\mathcal{O}_L/\mathfrak{P}_i$ is a degree $f$ extension of $\mathcal{O}_K/\mathfrak{p}$ , which in turn is a finite extension of $\mathbb{F}_p$ (where $\mathfrak{p}\cap\mathbb{Z}=(p)$ ).

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number fields

Decomposition and Inertia Fields

Galois theory is an important tool for investigating the factorization of primes in number fields. The topic at hand is how certain subgroups of the Galois group and intermediate fields encode factorization in Galois extensions, and how these can be used to prove facts about general extensions or transfer them to the Galois case. I learned this stuff from the books Number Fields by Marcus (thanks to Alex Youcis for the recommendation) and Algebraic Number Theory by Neukirch.

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schemes

Normalization of Algebraic and Arithmetic Curves

Normal schemes are nice, and happily there is a process for taking a scheme and producing “the same scheme but normal”, namely normalization. This can be thought of as a mild analogue of resolution of singularities (whose goal is to produce “the same scheme but nonsingular”), and indeed in the case of curves normalization succeeds in resolving singularities.

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galois representations

Eichler-Shimura Motivation and Overview

This semester (Spring 2016) I’m undertaking a study of Eichler-Shimura, i.e. associating Galois representations to modular forms. My vision is to be thorough and learn all the theory that gets involved from algebraic geometry and number theory and modular forms. This note will give some motivation and an overview of some of the things that will be coming up in this study. There’s some great motivation in Jared Weinstein’s paper Reciprocity laws and Galois representations: recent breakthroughs, which is where I’m getting a lot of this. (Thanks are due also to Alex Youcis for comments on the original.)

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