# Valuation rings and specialization

Valuation rings can be defined in a variety of equivalent ways. They are integral domains:

1. with a valuation to a totally ordered abelian group.
2. where the set of ideals is totally ordered with respect to inclusion.
3. where for every $x$ in the fraction field, either $x$ or $x^{-1}$ (or both) lies in the ring.
4. They are local rings that are maximal under the notion of domination with respect to local rings contained in a fixed field. We say that an inclusion $f: A \to B$ is a domination if the maximal ideal of $B$ pulls back to that of $A$.

It is a simple and useful exercise to show that these are equivalent and the stacks project does a fine job of it.  Personally, I find it easiest and most useful to use definitions 1+2+3. The main lemma about valuation rings is:

Main Lemma: Suppose $A \subset K$ is a local ring contained in a field. Then there exists a valuation ring $V \subset K$ that dominates $A$.

Before we move on to the proof, let me briefly explain why it’s important. This lemma is more or less equivalent to saying that on any scheme $X$, a specialization $x \to y$ can be realized by a map $\mathrm{Spec} V \to X$ with $V$ a valuation ring whose maximal ideal maps to $y$ and generic point maps to $x$.

This is, for instance, how valuation rings show up in the valuative criterion of properness or the arc topology of Bhargav-Mathew.

# Semistable Abelian Varieties and unipotent action

This blog post is inspired by Ribet’s “Endomorphisms of semi-stable abelian varieties over number fields”. I will talk about semi-stability for Abelian varieties, the minimal field of definition of endomorphisms and Grothendieck’s quasi-unipotent theorem.

Let’s begin with semi-stable Abelian varieties. Suppose we have a DVR $R$ and an Abelian variety over $K$, the fraction field of the DVR. We say that the Abelian variety has good reduction if there is a smooth, proper model over $R$. On the other hand, we say that an Abelian variety has semi-stable reduction if the special fiber of the Neron model is the extension of an Abelian variety by an algebraic torus.

By Chevalley’s theorem, this is equivalent to saying that there is no additive portion over the special fiber. In the case of elliptic curves, semi stable means that the reduction can be either an elliptic curve or $\mathbb G_m$. We have the following criterion for semi-stability:

Theorem 1: In the above situation, an Abelian variety has semi-stable reduction if and only if the action of the inertia group of $K$ on the Tate module is unipotent. Continue reading

# The reduction of base change theorems to nice cases

There are various base change theorems in algebraic geometry (flat in zariski topology, proper, smooth in etale topology etc). Often, especially in the etale site, it is easier to prove base change for specific morphisms (relative dimension one or projective instead of proper, for instance) and there is a formal mechanism for reducing from the generic case to the specific case. This post sketches a proof of that.

# Why did Iwasawa work so hard?

This post is very rough and is only to remind me of my chain of thought. Read at your own peril.

The first example of Iwasawa theory most people see is the historically first one of class groups for the cyclotomic $latex\mathbb Z_p$ tower of $\mathbb Q$. The reason the idea works at all is that somehow, passing to the limit reduces the “amount of bad stuff” that can happen and the key to this is a sort of control theorem where, if we denote the inverse limit of class groups by $X$ which has an action of $\Lambda \cong \mathbb Z_p[[t]]$, then Iwasawa shows that the class groups $X_n$ at finite levels can be recovered by essentially doing $X/v_n X$ where $v_n = (1+t)^{p^n} - 1$. This step very crucially uses class field theory in the identification of the class group with the maximal unramified extension.

Now one might object that perhaps we don’t need to work so hard. After all, $X = \lim X_n$ where the limit is over the algebras $\mathbb Z_p[[t]]/v_n \cong \mathbb Z_p[\theta]/(\theta^{p^n} - 1)$ where $\theta = 1+t$ and it seems natural that $X/v_n X = X_n$. Unfortunately (or fortunately), this is not true at all and is where Iwasawa theory gets it’s great power from. The purpose of this post is to record how bad things can go without any control theorems.

# Solving non linear differential equations using combinatorics!

In this post, we will consider the (nasty?) non linear differential equation:

$t(h'(t))^2 - 2th'(t) + 2h(t) = 0$

and show that the power series:

$h(t) = \sum_{n\geq 0}\frac{n^{n-2}t^n}{n!}$

is a solution using combinatorics! In particular, note that $h(t)$ is the exponential generating function for the number of labelled trees on $n$ vertices by Cayley’s formula. Let us call this number $T_n$, which is equal to $n^{n-2}$ by Cayley.

If we plug in the generating function into the differential equation, we find that in order to show that $h(t)$ is a solution to the differential equation, it is sufficient to show the following recurrence:

$2(n-1)T_n = \sum_{k=}^{n-1}k(n-k)T_kT_{n-k}{n \choose k}$

We will show this recurrence directly using combinatorics:

We will count the number of labelled trees on $n$ vertices with a distinguished edge with a distinguished direction. This number is clearly $2(n-1)T_n$ since there are $n-1$ edges to choose from and $2$ directions.

But if we imagine deleting the edge, this is the same as taking two labelled trees of sizes $k, n-k$ with a distinguished vertex on each and a choice of $k$ elements from $n$. The choice of $k$ elements is because once we have two trees with labels from $\{1,\dots, k\}$ and $\{1,\dots, n-k\}$ we need to map the union of these two sets into $\{1,\dots, n\}$ and once we pick a $k$ element subset of $\{1,\dots, n\}$, we can map the first set into it, preserving order and similarly for the complement.

Then, on a tree of size $k$, there are $k$ distinct vertices to choose from.

This correspondence is clearly bijective and so the number we are counting is also equal to:

$\sum_{k}k(n-k)T_kT_{n-k}{n\choose k}.$

as promised.

I do not know how to prove the recurrence directly! But there are many nice proofs of Cayley’s theorem available. One I particularly like is here.

# Obstructions to Weil Cohomology coming from supersingular elliptic curves and an extension to ordinary elliptic curves

When Serre and Grothendieck were coming up with an extension of the usual cohomology theory to varieties in algebraic geometry, an important example was given by Serre which showed that you couldn’t have a Weil cohomology theory with coefficients in $\mathbb Q_p$ in characteristic $p$. I will explain the example and extend it to show something about the ordinary case.

# Constructible sets, openness of flat maps and generic freeness.

I will cover a set of results that are connected and illustrate some interesting techniques about working with finiteness conditions (finitely generated as a ring, module etc) in algebraic geometry over a field.

Initially, I will prove Chevalley’s theorem that the image of a constructible set is constructible. This will involve proving Grothendieck’s generic freeness and Noetherian induction. Next, I will give a couple of applications of Chevalley’s theorem – I will prove that flat maps are open and provide a different proof of the Nullstellensatz.

# Irreducibility of Galois representations attached to Elliptic Curves

Epistemic status: I have not checked this carefully for errors, it is entirely possible there are mistakes in this.

We will be mostly be concerned with Elliptic curves over number fields $E/K$. However, let us start with a general lemma about Elliptic curves with endomorphism ring $\mathbb Z$. In this post, I collect some properties about representations coming from Elliptic curves.

# Weak Mordell-Weil and various approaches to it

I will show a few different approaches to the Weak Mordell-Weil theorem and how they are all really the same proof. The various proofs use the Hermite-Minkowski theorem, Class Field Theory and the Dirichlet’s Unit theorem/finiteness of Class group.

Let $E/K$ be an elliptic curve over a number field. The Mordell-Weil theorem says that the group of rational points $E(K)$ is finitely generated. This is usually proven in two steps:

1. Weak Mordell-Weil Theorem: We prove for some $n$ that $E(K)/nE(K)$ is finite.
2. Theory of heights: We define the notion of a height of a point on $E$ (roughly, how many bits of information one would need to store the point). Using this and (1), the completion of the proof is quite formal.

I will focus here on the weak Mordell-Weil theorem and in particular, an approach to it using the Hermite-Minkowski theorem. This approach will apply without very little change to the case of Abelian varieties and the general technique seems to be applicable in great generality.

The idea of the proof is as follows:

1. Reduce the case of general $K$ to supposing that $K$ contains the n-torsion using the Kummer sequence.
2. Use the Kummer Pairing to reduce to showing that the inverse image of $E(K)$ under the multiplication by $n$ $[n]$ map generates a finite extension of $K$.
3. Show that there is a smooth, proper model of $E/K$ over an open subset of $\mathcal O_K$ and hence the inverse image of $[n]$ generates an extension etale over $R$, not just $K$.
4. Apply Hermite-Minkowski.

# Weil pairing and Galois descent

There is an interesting way in which Weil pairing on Abelian Varieties is nothing more than Galois descent for a particular Galois extension. I have not seen this connection used before in the literature but I have also not seen a lot of literature…

Let $A/k$ be an abelian variety over a field $k$ of characteristic $p$. Suppose $m$ is an integer coprime to $p$ and let $[m]: A \to A$ denote the multiplication by $m$ map with kernel $A[m]$.

Recall that there is a dual abelian variety $A^\vee$ representing the Picard functor for $A$. In particular, $A^\vee[m]$ is the group of line bundles $\mathcal L$ on $A$ such that $[m]^*\mathcal L$ is trivial. $A$ and $A^\vee$ are finite abelian groups and the Weil pairing is a perfect pairing of the form:

$\langle-,-\rangle: A[m] \times A^\vee[m] \to \mu_m.$

It’s definition goes as follows: Let $D$ be a divisor corresponding to $\mathcal L$ and let $g(x)$ be a rational function on $A$ such that $\mathrm{div}\ g(x) = [m]^*D$. Then, for $a \in A[m]$, we define $\langle a,\mathcal L\rangle = g(x+a)/g(x)$.

If $t_a$ denotes translation by $a$, then $g(x+a)$ is the pull back of $g(x)$ along $t_a$. Since $[m]\circ t_a = [m]$, $g(x+a)$ and $g(x)$ have the same divisor and so $g(x+a)/g(x)$ is regular everywhere on $A$ and hence constant by properness of $A$. A little more work shows that this rational functions is in $\mu_m$. We will see below that this is an immediate conclusion of our alternate viewpoint.

This is the standard picture. However, there is also this alternate way of looking at things:

Consider the etale Galois extension $[m]: A \to A$. It’s galois group is canonically identified with $A[m]$. Moreover, $A^\vee[m]$ is precisely the set of line bundles on $A$ trivialized by this etale cover. In other words, it is the set of Galois twists of $\mathcal O_A$ (or even for any line bundle). Since descent is effective for this extension, this will immediately imply that:

$A^\vee[m] \cong H^1(\mathrm{Gal}([m]), \mathcal O_A^\times) = H^1(A([m]), k^\times) = \mathrm{Hom}\ (A[m],k^\times).$

The final equality is because $A[m]$ acts trivially on $\mathcal O_A^\times = k^\times$. We also see immediately that $\mathrm{Hom}\ (A[m],k^\times) = \mathrm{Hom}\ (A[m],\mu_m)$ since $A[m]$ is m-torison as an Abelian group. Therefore, this gives us a pairing:

$A[m]\times A^\vee[m] \to \mu_m.$

Explicitly, the map goes as follows: Given a line bundle $\mathcal L$, we pick an isomorphism $g: O_A \to [m]^*\mathcal L$. Then, the corresponding 1-cocycle for $a \in A[m]$ is defined by $a \to (t_a^*g)^{-1}g \in \mathrm{Aut}(\mathcal O_A)$. It is easily seen that this is the same explicit construction as in the standard viewpoint.

#### Conclusion:

I find the Galois descent viewpoint conceptually satisfying. The standard treatments of the Weil pairing can seem arbitrary and it not clear why such a pairing should exist or be useful. On the other hand, $[m]:A \to A$ is a perfectly natural Galois extension connected to $A$ and $\mathcal O_A$ torsors are clearly an interesting thing to consider.