The Height of a Formal Group Law in terms of the Symmetry of the Underlying CM Abelian Variety

This proof was made possible by a couple helpful and fabulous conversations with Yifeng Liu. All errors are mine and mine alone.

This is toward my understanding of the phrase “Why is height so important as an invariant? Because the height of a formal group law comes from the symmetry of the underlying variety.”

One method of getting lower dimensional formal group laws from abelian varieties of higher dimension is via using the theory of complex multiplication — splitting the abelian variety by splitting the prime (as I exposited in my paper here).

I show that for abelian varieties with CM, the height of the formal group law pieces are expressible as a formula in terms of the degree of some field extensions of $Q_p$ (one corresponding to each prime living over $p$) and the dimension of the rational endomorphism ring of the variety as a $Q$-vector space.

Notation:

• $F$ is a formal group associated to a CM abelian variety $A$.
• $\pi = \pi_A$, the geometric Frobenius of $A$
• $L := Q(\pi)$ with $[L : \mathbb{Q}] = e$
• $D := End^0(A)$ with $[D: L] = r^2$
• $dim(A) = g = er/2$.
• $L$ is the center of $D$
• Assume that $\mathcal{O}_L \subset End(A)$ (which we may after replacing $A$ by an isogenous abelian variety).
• Consider the set $\Sigma^{(p)}_L$ of discrete valuations of $L$ dividing the rational prime number $p$.

Theorem:

1. the decomposition $$D \otimes \mathbb{Q}_p = \prod_{w \in \Sigma^{(p)}_L} D_w$$ and $$\mathcal{O}_L = \prod \mathcal{O}_{L_w}$$
gives a decomposition $$F = \prod_w F_w$$
2. The height of $F_w$ equals $[L_w : \mathbb{Q}_p]\cdot$ r.

Die Philosophie der formalen Gruppengesetzen der elliptischen Kurve

Ich lerne Deutsch. Bitte, vergizieh mir und meine Unwissenheit der deutschen Grammatik.

In der Studie von Gruppen mit topogischen Struktur, wir haben die globale Objekt (die Gruppe) mit eine lokalen Objekt (die infinitesimale Gruppe) ersetzen. Wir betrachten dieses Spiel folgt vor:

1. Wir beginnen mit eine Raum.
2. Wir definieren einem binäre Operation an der Menge von Punkten in diesem Raum (einem Operation ist kommutativ, assoziativ, unital, und hat Inversen)
3. Wir leiten eine infinitesimale Gruppe.

z.B. Von Lie-Gruppe (Gruppe internen in die Kategorie der glatten Mannigfaltigkeiten) nach eine Lie-Algebra (Gruppen internen in die Kategorie der unendlich Mannigfaltigkeiten).

1. Wir beginnen mit einer glatten Mannigfaltigkeit (von Geschlect 0 oder 1).
2. Wir ein Produkt Morphismus definieren (Lie-Gruppe).
3. Wir leiten eine infinitesimale Gruppe  (Lie-Algebra).

Group Law on the (Punctured) Affine Line

There are likely inaccuracies in this post, as I am just beginning to learn the basics of algebraic geometry. Constructive criticism is strongly encouraged.

There once was a line…

Let’s look at the affine line over $\mathbb{C}$. This is just the complex line with no distinguished element (i.e., a plane which forgot it’s origin — it’s 2 real dimensions, or, equivalently, 1 complex dimension).

$\mathbb{A}^1 \simeq \text{Spec } \mathbb{C}[x]$

As we know, the affine line over the field $\mathbb{K}$ is isomorphic to the spectrum of a ring of single variable polynomials (with coefficients in $\mathbb{K}$). If you aren’t familiar with this isomorphism, I recommend popping over to Spectrum of a Ring. For simplicity, let’s work with the field $\mathbb{C}$, although, I’m pretty sure the rest of this post still works for any $\mathbb{K}$.

Is there a reasonable way to take in two points, and ask for a third?

This is generally a fun question to answer when you’re handed a space. So, how do we add two points of $\mathbb{A}^1$ to get a third point in $\mathbb{A}^1$? Continue reading Group Law on the (Punctured) Affine Line

$Pic(X)$ vs. $CP^\infty$

Thank you to Edward Frenkel for kindly explaining the difference between $CP^\infty$ and $Pic(X)$ (both classifying spaces of line bundles), and to Qiaochu Yuan for explaining why on earth $CP^\infty$ is the moduli space of line bundles over a point. Any errors are mine, not theirs.

As we saw in a Precursor to Characteristic Classes, $CP^\infty$ is the classifying space of complex line bundles over $X$.

$CP^\infty$ is, in some sense, the moduli space of line bundles over a point. There’s only one isomorphism class of line bundles over a point — but then this one line bundle has automorphism group $C^\times$ (which is homotopy equivalent to $U(1)$).

Allow me to introduce you to something that looks a LOT like $CP^\infty$. Continue reading $Pic(X)$ vs. $CP^\infty$

Spectrum of a Ring

Approaching Scheme Theory as a Beginner

Thank you to James Tao for explaining the following, geometrically intuitive, interpretation to me.

A scheme is the result of patching affine schemes together, an affine scheme is any locally ringed space that is equivalent to Spec of a ring. To make an analogy, a manifold is the result of patching together copies of $\mathbb{R}^n$, in a way that preserves the smooth structure.

And the smooth structure of $\mathbb{R}^n$ is essentially the fact that it is a sheaf: over any open set, we have the space of smooth functions on that set, and these functions restrict and glue nicely, etc.

So when you go from $\mathbb{R}^n$ to “smooth manifolds,” you are creating spaces that locally look like $\mathbb{R}^n$, respect the same kind of smooth structure (as captured by the fact that smooth functions are defined on open sets in a way independent of the patching), yet have some global structure that might prevent them from just being the same as $\mathbb{R}^n$.

If you care about algebraic (polynomial / rational) functions defined on algebraic sets (zeroes of systems of polynomials), then you can do the same program! The “basic patch” is “affine scheme,” and you patch affine schemes together to get a scheme. Continue reading Spectrum of a Ring