Month: May 2015

This post assumes familiarity with the Landweber exact functor theorem, elliptic genera, and bordism theories.

An ongoing desire of mine is to geometrically approach elliptic spectra.

Note that I’m not talking about geometric cocycles for tmf. I’m talking about the vague goal of understanding elliptic spectra in their own right using “geometric” techniques (properties of the object that are invariant under a chosen collection of transformations of that object).

There is a presentation of an elliptic homology theory as a bordism theory with singularities outlined by Landweber in Elliptic Cohomology and Modular Forms (based on this paper which presents \(H_*(-;\mathbb{Z})\) as a bordism theory with singularities).

This seemed like the beginning of the answer to my desire, but I now think that this construction is basically a less intuitive version of \(MSO_*(-) \otimes_{MSO_*} R\), thinking about “tensoring out” classes in \(MSO\) as coning them off. I’ll explain what I mean by “coning them off” in a bit, first let me outline the construction:

Outline:

1. Start with an elliptic genus \(\pi_*(MSO) \xrightarrow{\phi} R\)
2. Mod out the ring spectrum \(MSO\) by \(ker(\phi)\) to get a spectrum “\(MSO/ker(\phi)\)” whose homotopy groups are \(R\)
   i.e., construct \(F: \pi_*(MSO/ker(\phi)) \to R\) s.t. \(ker(F) = 0\)
3. Check that the spectrum \(MSO/ker(\phi)\) is a ring spectrum

Let’s go through it!

Before we begin, you might wonder why we’re working with \(MSO\). Landweber works with \(MSO_*[\frac{1}{2}]\) and not \(MU\); the map from \(MU\) to our elliptic spectrum \(E\) factors through \(MSO\) (we can forget structure on our bordisms).

Notational side, \(MSO_* := \pi_*(MSO)\), and \(\mathbb{S}^n\) means the the suspension spectrum with \(S^n\) as it’s 0th space, i.e. \(\Sigma^\infty S^n_+\).

Reason to kill the generators

Start with an elliptic genus, a concrete ring homomorphism \(MSO_* to R\) that satisfies some properties. Let’s look at \(R := \mathbb{Z}[\frac{1}{2}][\delta, \epsilon]\).

What generators of \(MSO_*\) are killed by this elliptic genus?

In other words, what is \(ker(MSO_* \xrightarrow{\phi} \mathbb{Z}[\frac{1}{2}][\delta, \epsilon])\)? Let’s say \(\delta\) has degree 2, and \(\epsilon\) has degree 4.

As we know (thanks Milnor), \(MSO_*[\frac{1}{2}] \simeq \mathbb{Z}[x_2, x_4, x_6, x_8, …]\) (\(MSO_*\) has some 2-torsion, adjoining 2 allows us to think of \(x_i\) as \(CP^{i}\)) so we can rewrite \(\phi\) as:

\\(\mathbb{Z}[\frac{1}{2}][x_2, x_4, x_6, x_8, …] \xrightarrow{\phi} \mathbb{Z}[\frac{1}{2}][\delta, \epsilon]\\)

\(\mathbb{Z}[\frac{1}{2}][x_2, x_4, x_6, x_8, …]/(x_6, x_8, …) \simeq Z[\frac{1}{2}][x_2, x_4]\)

We require that mult-by-\(\alpha\) acts injectively in \(E\) s.t. \(C(\alpha)\) is just the cokernel of \(\times \alpha\).

If mult-by-\(\alpha\) acts injectively, then \(0 \to \pi_k(\Sigma^E) \xrightarrow{\times \alpha} \pi_k(E) \to \pi_k(C(\alpha)) \to 0\) is an exact sequence for all \(k\), and \(\pi_*(C(\alpha))\) will give us the ring we’d expect, \(\pi_*(E)/\alpha\).

\\(MSO_*/(x_6, x_8, …) \simeq R\\)

But can we do this on the level of spectra? Can we construct \(MSO/(x_6, x_8, …) \simeq E\) s.t. \(\pi_*(E) \simeq \mathbb{Z}[x_2, x_4]\) by coning off unwanted generators?

Well, before we get caught up in fantasy, what does “\(MSO/\alpha\)” even mean?

Understanding the quotient

Let’s use our intuition for what \(R/\alpha\) should be when \(R\) is a ring, that is, the exact sequence \(0 \to (a) \to R \to R/\alpha \to 0\). I point this out to motivate that the quotient of a ring spectra by the ideal of a ring will be a cokernel (in nice cases).

Given an element \(\alpha \in \pi_n(E)\), we get a map \(\Sigma^n M \xrightarrow{\times \alpha} M\) where \(M\) is an \(E\)-module.

How? We have an element \(\alpha \in \pi_n(E)]\), that is:

\\(\mathbb{S}^n \xrightarrow{\alpha} E\\), apply the functor \(- \wedge M\)

\\(\mathbb{S}^n \wedge M \to E \wedge M\\)

Note that a ring spectrum \(M\) is a monoid object in the category of spectra (which is a monoidal category), we have \(M \wedge M \to M \leftarrow \mathbb{S}\), with associativity and identity. Since \(M\) is an \(E\)-module, we have the monoid action \(E \wedge M \to M\), so we compose and get:

\\(S^n \wedge M \to E \wedge M \to M\\)

We return our attention to the map: \(\Sigma^n M \xrightarrow{\times \alpha} M\)

Take the mapping cone of \(\times \alpha\):

\\(\Sigma^nE \xrightarrow{\times \alpha} E \to C(\alpha)\\)

Taking homotopy groups:

\\(… \to \pi_k(\Sigma^nE) \to \pi_k(E) \to \pi_k(C(\alpha)) \to \pi_{k+1}(\Sigma^nE) \to …\\)

We require that mult-by-\(\alpha\) acts injectively in \(E\) s.t. \(C(\alpha)\) is just the cokernel of \(\times \alpha\). If mult-by-\(\alpha\) acts injectively, then \(0 \to \pi_k(\Sigma^E) \xrightarrow{\times \alpha} \pi_k(E) \to \pi_k(C(\alpha)) \to 0\) is an exact sequence for all \(k\), and \(\pi_*(C(\alpha))\) will give us the ring we’d expect, \(\pi_*(E)/\alpha\)).

Conceptually (and formally), we might think of the members of \(ker(\phi)\) as stratifolds.

Coning off the members of the kernel (i.e., killing the appropriate generators of \(MSO\))

So, back to the main story: killing the appropriate generators of \(MSO_*[\frac{1}{2}] \simeq \mathbb{Z}[\frac{1}{2}][x_2, x_4, x_6, x_8, …]\) until we get \(\mathbb{Z}[\frac{1}{2}][x_4, x_6]\) — but now we’re pulling this on the level of spectra. We now know that when we say “\(MSO/x_6\)” what we mean is:

We’re looking at \(x_6 \in \pi_n(MSO)[\frac{1}{2}]\), and we take the mapping cone: \(\Sigma^nMSO \xrightarrow{\times x_6} MSO \to C(x_6) =: MSO/x_6\)

We can keep going in this vein, next looking at the sequence \(\Sigma^n C(x_6) \xrightarrow{\times x_8} C(x_6) \to C(x_6, x_8)\), and iterate: say \(x_i \in \pi_t(MSO)\), we’ll eventually come to:

\(\Sigma^t C(x_6, …, x_{i-2}) \xrightarrow{\times x_i} C(x_6, …, x_{i-2}) \to C(x_6, x_8, …, x_i)\)

We’ve constructed the map \(MSO \to C(x_6, …, x_i)\) — requiring that the ideal \((x_6, …, x_i)\) acts injectively (s.t. the mapping cone is just the cokernel of the map).

Taking the homotopy groups of \(C(x_6, …, x_i)\), we find \(\pi_*(C(x_6, …, x_i)) = \mathbb{Z}[x_2, x_4]\).

We know that \(C(x_6, …, x_i)\) is a spectrum, but is it still a ring spectrum? If the genus is Landweber-exact (i.e., tensoring out \(x_6, …, x_i\) gives us a multiplicative homology theory), the resulting spectrum \(MSO/(x_6, …, x_i)\) a ring spectrum.

This feels a bit hacky, but I’m not sure how else to prove that \(C(x_6, …, x_i)\) is a ring spectrum besides some messy obstruction theory which I know nothing about.

I mentioned at the beginning that this construction is less intuitive than tensoring the underlying coefficient ring as an \(MSO_*-module\), this is because we don’t have an obvious criterion for when the resultant spectrum is a ring spectrum!

Some remaining uncertainties

Let’s say we wanted to “geometrically” interpret inverting 2, even though this is a very algebraic thing, in the case of \(MSO[\frac{1}{2}]\)? I’m not quite sure how to do this precisely — the best I can currently do is outline a picture. Recall that \(MU_n(X) := {M^n \to X/ \sim}\), which is the bordism classes of maps to X from an n-dimensional stably almost-complex manifold, so the \(\times 2\) map is then the disjoint union of bordant classes of manifolds. Perhaps this gives us a way to invert \(2\) geometrically? I’m really not sure how to make this precise.

Assuming that all of this works, this bordism with singularities \(C(x_6, …, x_i) \simeq E\) is a presentation of the spectrum associated to our elliptic genus.

If we instead start with an elliptic genus \(MU_* \to R\), can we get the same result? How does the grading differ or is it the same because we are factoring through \(MSO\)?

As a last comment, I’ll address my vague desire for the geometry of elliptic curves to come into this story. Looking, say, at the 2-torsion points of our elliptic curves — if this construction is functorial — we might expect a 2-torsion action on the bordism with singularities presentation. This might be close to what “tmf with level-2 structure” means, but I’m really not sure.

Thanks to Chris Sommer-Preis, Achim Krause, and Tomer Schlank for answering some of my questions on the topic.

Eine deutsche Übersetzung des folgenden Abschnitts befindet sich hier.

Philosophical Motivation

In the study of groups with topological structure, we commonly replace the global object (the group) with a local object (the infinitesimal group). We play the following game.

1. Start with a space.
2. Define a method of adding two points to get a third point of this space (which is associative, unital, commutative, and has inverses).
3. Derive an infinitesimal group.

Examples:

• Lie group (group internal to the category of smooth manifolds)
  \(\to\) Lie algebra (group internal to the category of infinitesimal spaces).
• elliptic curve formal group
  \(\to\) elliptic curve formal group law.

1. Start with an elliptic curve over a scheme \(\text{Spec }R\).
2. An elliptic curve is a group scheme (over \(\text{Spec }R\)) whose underlying object is 1-dimensional, proper, and smooth.
   In less abstract terms, it comes with an addition law and a marked point for free.
3. Formally complete the elliptic curve along the origin
   In less abstract terms, take the Taylor series expansion of the addition law.

The phrase “formal completion of E along 0″ looks like the Taylor series expansion of an elliptic curve about the origin. In some sense, ‘completion at a point’ does give you the Taylor series of a map in the algebraic-geometric setting, but it’s more general than this. That is, a Taylor series is usually of the form \(f(x+y) = f(x) + yP_1(x) + y^2P_2(X) + …\), where \(P_i(x) = \frac{f^{(i)}(x)}{i!}\).

For more on this, Aaron Mazel-Gee wrote a great paper on Dieudonne modules.

We can view formal completions as key to our search for an algebraic version of a Lie Algebra. In differentiable Lie theory, the Baker-Campbell-Hausdorff theorem describes the group-law in a small neighborhood of the identity. However, this uses the exponential, which fails to make sense, even as a formal power series, in characteristic \(p\).

This justifies our quest: describe the group law in a small neighborhood of the identity of the curve without using the exponential map. First, we must go on an excursion to understand completion.

You Complete Me

Recall the method of obtaining \(\mathbb{Z}_p\), the p-adic completion of the integers.
\begin{align*}
& \mathbb{Z}/p \hookleftarrow \mathbb{Z}/{p^2} \hookleftarrow \mathbb{Z}/p^3 … \\
\mathbb{Z}_p & = \text{lim } \mathbb{Z}/p^n
\end{align*}

Let \(R\) be a ring. We have a polynomial ring, \(R[t]\). How do we make a ring of formal power series from this? Well, note that \(R[t]/t = R\), so we get:
\begin{align*}
& R \hookleftarrow R[t]/t^2 \hookleftarrow R[t]/t^3 \hookleftarrow … \\
R[[t]] & = \text{lim } R[t]/t^n
\end{align*}

Similarly,
\begin{align*}
& \text{Spec }A \hookrightarrow \text{Spec }A[t]/{t^2} \hookrightarrow \text{Spec }A[t]/t^3 \hookrightarrow … \\
\text{Spf }A[[t]] & = \text{colim }\text{Spec }A[t]/t^n
\end{align*}

Completion is the process of picking a ring \(R\) together with a maximal ideal \(m\) and forming the limit of \(R/m^n\).

As an example, we consider the group scheme \(E\) (for example, an elliptic curve) over \(\text{Spec }A\).

\(\text{Spec }A[t]/t^2\) is the infinitesimal neighborhood of the first derivative/first infinitesimal neighborhood of \(\text{Spec }A\). This is like truncating the Taylor series after the information given by the first derivative, e.g. Lie algebra. Similarly, \(\text{Spec }A[t]/t^3\) tells us about 2nd derivative, and we need a larger infinitesimal neighborhood for the second derivative. In the case of a formal group law, we want to write down all of the infinitesimals, so we take the colimit of \(\text{Spec }A[t]/t^n\) and get a formal scheme.

Zariski locally on \(S\), the formal completion \(\hat{E}\) of \(E\) along the zero section is of the form:

\\(\hat{E} \simeq \text{Spf }(A[[t]])\\) where \(\text{Spf }A[[t]] := \text{colim }\text{Spec }A[t]/t^n\)

For example, Zariski-locally over \(Spec(A)\), we can put an elliptic curve in Weierstraß normal form \(y^2 + a_1 xy = x^3 + a_2 x^2 + a_4 x + a_6\) (I think outside of characteristic 2 and 3, we may take \(a_1 = 0\)).

In this form, the variable \(z = x/y\) is a uniformizer at the identity (\(y\) has a pole of order 3 at the identity, \(x\) has one of order 2, so \(x/y\) has zero of order 1). It’s reasonable to expect \(Spf(A[[z]])\) to be the formal completion.

Note that we are choosing an isomorphism \(G \simeq \text{Spf }R[[t]]\) here, that is, choosing where to send \(t\) in \(G\) (sidenote: if \(G\) is associated to a cohomology theory, our choice of where to send \(t\) corresponds to a choice of complex orientation for the cohomology theory).

It is worth noting the difference between \(\varinjlim \text{Spec }A[t]/t^n\) and \(\text{Spec}\varinjlim A[t]/t^n\).
\\(\varinjlim \text{Spec }A[t]/t^n =: \text{Spf }A[[t]]\\)
\\(\text{Spec} \varinjlim A[t]/t^n =: \text{Spec }A[[t]]\\)
Formal schemes live in a category of limits of schemes (affine formal schemes live in a category of rings with the \(I\)-adic topology for some ideal \(I\)). So Spec(\(A[[t]]\)), considered as a formal scheme, would just be the trivial limit of Spec of the non-topologized ring.

Think about it: with Zariski topology, Spec(\(k[[t]]\)) is dense in Spec(\(k[t]\)), while Spec(\(k[t]/(t^n)\)) is not, for any \(n > 0\).

Our elliptic curve is a projective beast: reviewing the necessary coordinate changes.

Projective polynomials are defined by the following property: \(P(\lambda x, \lambda y, \lambda z) = \lambda^k P(x,y,z)\)

We can make any polynomial into a polynomial with this property by homogenizing (e.g. \(y^2z = x^3 + axz^2 + bz^3\)).

We ask for the completion of the elliptic curve at the identity wrt the group structure. That is, at the “point at infinity” if we write it in Weierstraß form. “Point at infinity” is the intuition, but “homogenize and look at the projective variety” is how its formalized. (Affine things are slightly awkward, and we can always homogenize equations by adding in another variable, so there’s a canonical way to compactify. Sadly, that’s often implicit in the literature.)

1. Input elliptic curve in homogeneous coordinates (\(y^2z = …\))
2. Coordinate transform (1) s.t. our elliptic curve contains the origin. (\(f(t) = t…\))
3. We want an equation for \(S\) in terms of \(T\), so we find the fixed point \(T = \phi(T)\) of (2) by repeatedly substituting \(t…\) for \(f(t)\). This gives us a power series of in terms of \(T\).

Let’s talk about step 2: we can’t talk about the point at infinity in our staring chart of the elliptic curve since it doesn’t include it , that is, \((0,0)\) doesn’t even generally lie on \(y^2 = x^3+ax+b\) so we must coordinate change to an affine coordinate CONTAINING that point.

There’s three maps from projective coordinates back to affine coordinates. All are defined on a certain open subset of \(\mathbb{P}^2\) (namely the one where a certain coordinate doesn’t vanish), and the corresponding map gives a coordinate chart for that open subset).

For an elliptic curve in Weierstraß form, the \(z \neq 0\) chart doesn’t contain the identity of the elliptic curve, but the \(y \neq 0\) has the identity at \((0,0)\). We’re basically changing coordinates from the classical affine equation \((y^2=..)\), which lives on the \(z \neq 0 thing\), to the other one living on the \(y \neq 0\) thing.

Think about it as follows: the elliptic curve lives in \(\mathbb{P}^2\), but we can describe its intersection with each of the coordinate charts. That’ll of course be different equations. The \(z \neq 0\) and \(x \neq 0\) are not suitable to talk about the neighbourhood of the identity, because they simply don’t CONTAIN the identity.

Derivation (for simplified Weierstrass):

The explicit derivation of elliptic curve formal group law from an elliptic curve is performed with a series of coordinate changes:

By plugging the expression for \(f(T)\) into itself repeatedly, \(f(T)\) stabilizes to a power series with \(T\) as the only variable.

It is sometimes tiresome (and sometimes meditative) to go through the coordinate changes and explicit solving for \(f(T)\) in terms of \(T\) on paper. sage (available free online) automates this process:

sage: E = EllipticCurve([2,3]).formal_group(); E}
Formal Group associated to the Elliptic Curve defined by y^2 = x^3 + 2*x + 3 over Rational Field
sage: F = E.differential(15); F
1 + 4*t^4 + 9*t^6 + 24*t^8 + 120*t^{10} + 295*t^{12} + 1260*t^{14} + O(t^{15})

An afternote on Cartier’s curves

I’ll wrap this up by mentioning Cartier’s curves. Cartier’s method relies very heavily on the fact that a formal group law is the formal spectrum of a power series ring. The previous method relied heavily on the fact that a formal group law \(G\) is the inductive limit of the finite subgroup schemes \(ker(p^n)\), where \(p^n\) is the \(n\)th Frobenius morphism: \(G \to G\), \(x \mapsto x^{p^n}\). This relies on all modules being ind-finite, which Lubin mentions to be hopeless in the case of Dieudonne modules.

Let \(B\) be any commutative \(A\)-algebra which is separated and complete over the topology defined by \(I^n\) where \(I\) are ideals of \(B\) (i.e., let \(B\) be a formal \(A\)-scheme).

For example, take \(B = A[[t]]\), the formal power series ring in one variable. The ideal \(I = tA[[t]]\), the set of \(n\)-tuples of elements (type: power series) in \(I\) may be equipped with a composition law \(\textbf{F}(a, b) = a*b\) (this composition will be finite, as we are working with nilpotent elements). Cartier calls such maps \(\text{Spf }A[[t]] \xrightarrow{F} \text{Spf }A[[t]]\) “curves in \textbf{F}.”

This name choice makes sense. Recall that we can define a polynomial over \(\mathbb{C}\) as \(\mathbb{C}[t] \to \mathbb{C}[t]\), similarly, we can define a curve over over \(\text{Spec }A\) as \(\text{Spf }A[[t]] \to \text{Spf }A[[t]]\).

Thank you to Akhil Mathew (for explaining the definition of formal schemes), Jesse Silliman (for answering my various questions on completion and formal schemes), and Achim Krause (for walking me through the derivation of the group law).