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?

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 $K$.

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.

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.

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.

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.

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 $\mathfrak{p}\cap\mathbb{Z}=(p)$).