# 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.

# Noether Normalization, Spreading out and the Nullstellensatz .

Hilbert’s Nullstellensatz plays a central role in algebraic geometry. It can be seen as the fundamental link between the modern theory of schemes and the classical theory of algebraic varieties over fields. Since this is one of the first results a novice in algebraic geometry learns and is often proved very algebraically, one often does not gain a good understanding of the proof till much later.

I will present three proofs of the Nullstellensatz found in the literature from a geometric perspective. This will highlight the role of the “spreading out and specializing” common to the proofs that might not be obvious from an algebraic presentation. The last proof is a very short, self contained demonstration of the techniques. Along the way, we will also see a geometric proof of Noether Normalization.

The proof of Hilbert’s lemma is usually broken up into the following two steps: 1) Prove the weak Nullstellensatz and 2) Derive the strong Nullstellensatz using the Rabinowitsch or other means. I will be focusing solely on the first step in this post. Nothing in this is original except for the presentation.

# 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.