## 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.”

In short — high amount of symmetry in the underlying abelian variety implies a high height of its formal group law (the converse is NOT true, if this was true, Elkies’s supersingularity theorem would be false).

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.

## Newspaper Ad: Looking for a Variety

Hello, my name is Catherine. I don’t want much, just looking for a nice Variety to spend my days with. If you apply, I’d like you to have a well understood group law that comes from some 3-fold symmetry, but I’m a simple girl, and I don’t need your group law to be all fancy and closed — a group chunk (group law which closed at least locally to the origin) is fine by me. I’ll have to put you through an interview process to see if your group chunk gives me a formal group law which is height 3, but don’t worry, it’ll be painless. Please let me know if you have a friend that matches this description!

This post is mostly a set-up to an (ill-formed) question. It’s motivated by this question:

How do I construct a variety which gives me
a formal group law of height 3?

That is, I want it to have a nice kind of 3 fold symmetry which is reflected in it’s structure around a marked point.

Edit: I have found many such varieties with the help of Kato, which I talk about here! I am presently still working on using a n-tuply periodic Riemannian surface method (different than the one below) based on the work of Dami Lee.

I want this variety to not to be a bunch of copies of the additive or multiplicative group on $R^n$. I am trying to define an at least 3-dimensional variety with a group chunk (that is, a group law which is closed at least locally to the origin). I want this group chunk to not be isomorphic to an additive, multiplicative, or elliptic group, or products of such groups.

The previous variety I was looking at ended up being isomorphic to the additive group, though it was very pretty. Aaron and a few others derived a variety from the relationship between the lengths of the vertices to a point interior to an equilateral triangle, which I rederived with the help of Laurens. Unfortunately the group law Alex Mennen and I defined on the variety ended up being the additive formal group law. I didn’t recognize it at first because it had 2 layers of square roots as a disguise, but Jack Shotton pointed out that if we did a variable change to get rid of the square roots (a variable change I had been doing formally to make calculations easier) it became quite obviously isomorphic to the additive group.

It was also pointed out to me by Doug Ravenel that height 3 formal group laws cannot be dimension 2. For some reason to do with the symmetry of the Jacobian of a dimension 2 variety which I don’t understand. So, I look now to dimension 3. More specifically, I look at tetrahedrons — the analogue of the square lattice, in some sense.

We begin with the vague desire of deriving a variety from some relationship on a tetrahedron (hoping that this variety has both a group law, and that the group law is height 3). Inna suggested that I look at a right angled tetrahedron to make my life easier, so we will look there.

I had an idea for a group law on right-angled tetrahedrons: is a group in which each element is indexed by an angle, that is, the angle $\theta$ of the plane that intersects it symmetrically (to produce the tetrahedron).

Then, we might add their volumes. We now get a third volume. What is an angle which gives us a tetrahedron with this volume? Is it unique?

Inna and I talked about this and she referred me to a group law on angles of a tetrahedron, which looks multiplicative but involves $sin$, so perhaps is a bit more complicated. It looks something like this: $\alpha + \beta = \sin^2(\alpha)\sin^2(\beta)$. Where does this group law live? What variety has angles as points? Does the free abelian group generated by tetrahedra have a geometric structure we could use?

Another issue: We now have a group law, but no variety! The whole point was to define a group law close to the origin, but what is closeness in this group?

I stop writing with a question still quite ill-formed and fuzzy:

How might we derive a variety based on a tetrahedron which allows us to put an angle-y/volume-y group law on its points?

## What is the “universal enveloping algebra” of a formal group law?

I posted earlier a query toward exploring the analogy between

smooth algebraic groups over $\mathbb{R}$ or $\mathbb{C}$ ::  Lie algebras

smooth algebraic groups over R (any commutative ring) ::  Formal group law

in which I tried to answer this question, and ended up with “the Lazard ring doesn’t quite work,” which makes sense in retrospect, as the Lazard ring is not associated with any particular formal group law. When I say “formal group law” I mean “1-d formal group law.”

What is a formal group law? It’s an expression of the group structure of G in an infinitesimal neighborhood of the origin. At the Midwest Topology Seminar, talking with Paul V. and Dylan Wilson, I have a somewhat more satisfying answer.

What is a Lie algebra? It’s an expression of the group structure of $G$ at the FIRST infinitesimal neighborhood of the origin. In characteristic 0, this extends to a definition of the group structure in an infinitesimal neighborhood of the origin by the Baker Campell Hausdorff  formula.

Specifically, what is the universal enveloping algebra of a formal group law? It’s the formal group law itself! Well, more specifically:

Universal enveloping algebra :: Lie algebra

Functions on Formal group law  :: Formal group law

According to wikipedia, “The universal enveloping algebra of the free Lie algebra generated by X and Y is isomorphic to the algebra of all non-commuting polynomials in X and Y. In common with all universal enveloping algebras, it has a natural structure of a Hopf algebra, with a coproduct Δ. The ring S used above is just a completion of this Hopf algebra.”

A formal group law already has a Hopf algebra structure. This is just the cogroup on formal power series $R[[x]]$ induced by the formal group law, $f$, that is,

$$R[[x]] \to R[[x]] \widehat{\otimes} R[[x]]$$

$$x \mapsto f(1 \otimes x, x \otimes 1)$$

This is already complete! We’re a formal group law so we’re already completed at the origin! And, if we are a 1-d formal group law, we’re always commutative (unless our ring is nilpotent), so this is promising.

## A quick comparison of Lie algebras and formal group laws

This post assumes that you are familiar with the definition of Lie group/algebra, and that you are comfortable with the Lazard ring. Note: This is less of an expository post and more of an unfinished question.

Why care about formal group laws? Well, we want to study smooth algebraic groups, but Lie algebras fail us in characteristic p (for example, $\frac{d}{dx}(x^p) = 0$), so, rather than a tangent bundle, we take something closer to a jet bundle.

Lie algebras are to smooth groups over $\mathbb{R}$ or $\mathbb{C}$ as formal group laws are to smooth algebraic groups over any ring $R$.

I want to apply this analogy! I want this deeply. I’m trying to puzzle out how to see if this analogy is deep or superficial. How deep does the rabbit hole go? Let’s look at an example.

Given the universal enveloping algebra of a Lie algebra $\mathfrak{g}$, we might think of this as a deformation of the Symmetric algebra:

$$U_{\epsilon}(\mathfrak{g}) := T(\mathfrak{g})/(x \otimes y – y \otimes x – \epsilon[x, y])$$

$$\mathbb{C}[\mathfrak{g}^*] \simeq Symm[\mathfrak{g}] := T(\mathfrak{g})(x \otimes y – y \otimes x)$$

Action on all of this is the adjoint action, that is, the action which takes an element $g$ of a Lie group $G$ sends $X \mapsto gXg^{-1}$. Orbits of this action stratify the dual Lie algebra, and there is a symplectic form that lives on each orbit.

I want to think of the adjoint action as directly analogous to the compositional conjugation action on the spectrum of the Lazard ring (over a ring $R$). Continue reading A quick comparison of Lie algebras and formal group laws