My understanding is that a complex-oriented spectrum is a ring spectrum $E$ with a map $MU \to E$.

Analogously, a ring with a formal group law is a ring $R$ with a map $L \to R$ (where $L$ is the Lazard ring).

This analogy can be made explicit (assuming that $L \to R$ is Landweber-exact for all primes):

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Let $E$ be an elliptic spectrum. We know the ring spectrum $MU$, and thus our choice of elliptic curve determines $E$. In other words, the formal group law (associated to the completion of our elliptic curve about the origin) gives us a map $L \to R$, which determines $E^*(-) \simeq MU^*(-) \otimes_{L} R$.

Despite this relatively clear path from elliptic curve to cohomology theory, the following is frustratingly unclear to me! How does changing the elliptic curve actually affect what its associated cohomology theory detects?

More explicitly: What's an example of 2 elliptic curves over the same ring s.t. their associated cohomology theories detect different things?


2 Answers 2


Let me try to be very concrete. (This is more or less just an elucidation of part of Charles's answer.)

Consider the elliptic spectrum $TMF_0(2)$ with $\pi_*TMF_0(2) = \mathbb{Z}[\frac13][b_2,b_4,\Delta^{-1}]$. This is the modern name for the elliptic cohomology theory defined by Landweber, Ravenel and Stong; today one can define it as the $E_\infty$-ring spectrum $\mathcal{O}^{top}(\mathcal{M}_0(2))$ where $\mathcal{M}_0(2)$ is the moduli stack of elliptic curves with a level-$2$-structure.

For simplicity, let us localize everything in sight at the prime $3$. We want to complete at one point corresponding to a height $1$ elliptic curve over $\mathbb{F}_3$ and at one point corresponding to a height $2$ elliptic curve over $\mathbb{F}_3$.

An elliptic curve over $\mathbb{F}_3$ has height $2$ if and only if it has $j$-invariant $0$ in $\mathbb{F}_3$. (See either http://en.wikipedia.org/wiki/Supersingular_elliptic_curve or Hartshorne, Algebraic Geometry, 4.23.1.) The $j$-invariant of an elliptic curve $y^2 = x^3+ b_2x^2+b_4x$ is $(b_2^2-24b_4)/\Delta \equiv b^2/\Delta \mod 3$ where $\Delta = -8b_4^3$. Thus, the elliptic curves $$E_1: y^2 = x^3+x^2+x $$ and $$E_2: y^2 = x^3 + x$$ are of height $1$ and $2$ respectively. These elliptic curves correspond to the closed points $P_1$ and $P_2$ in $\mathcal{M}_0(2)$ defined by the homogeneous ideals $I_1 = (3, b_2-1, b_4-1)$ and $I_2 = (3, b_2, b_4-1)$ respectively.

The completions $R_1 = TMF_0(2)^\widehat{}_{I_1}$ and $R_2TMF_0(2)^\widehat{}_{I_2}$ have both homotopy groups isomorphic to $\mathbb{Z}_3[[x]][u, u^{-1}]$ for a formal coordinate $x$ at the points $P_1$ and $P_2$ respectively. But $R_1$ and $R_2$ are very different as $R_1$ has height $1$ and $R_2$ has height $2$. In particular, $v_1$ is invertible on $R_1$, but it is not on $R_2$ (because in the first case $v_1$ is non-zero modulo the maximal ideal, in the second case it is zero).

At the prime $3$, there is a $v_1$-self map on the Moore spectrum $S^0 /3$ (that is build as the cofiber of the map $S^0\xrightarrow{\cdot 3}S^0$. Denote the cofiber of this map $S^0/3 \xrightarrow{v_1} S^0/3$ by $X$. Then $(R_1)_*(X) = 0$, but $(R_2)_*(X) \not\cong 0$.

  • $\begingroup$ Thank you so much for the clarification! Let $E$ be a cohomology theory. Potentially stupid question: Is the statement "$v_n$ is invertible on $E$" equivalent to "$E$ detects $v_n$ periodic phenomena"? $\endgroup$ Apr 2, 2015 at 21:30
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    $\begingroup$ I think, the correct statement is that if $v_n$ is invertible on $E$, then $E$ does not detect $v_k$-periodic phenomena for $k>n$. For example, if you take $E= E(1)_{\mathbb{Q}}$ with homotopy groups $\mathbb{Q}[v_1^{\pm 1}]$, then $E$ does not detect $v_1$-periodic phenomena as the $v_1$-self map on the Moore spectrum $S^0/p$ induces $0$ on $E$. $\endgroup$ Apr 3, 2015 at 19:06

What does "detect different things" mean? I'll take it to mean "have different Adams-Novikov spectral sequences".

For Landweber exact theories $E$, the situation is pretty straight-forward (and not very exciting): the ANSS associated to $E$ depends only on the height of the theory (more precisely, on the heights at all primes).

For instance, suppose $E$ is Landweber exact, elliptic, and that $\pi_*E$ is $p$-local. Then $p$ is the only relevant prime, and the only possible heights are $0,1,2$, as these are the only possibly heights for the formal group of an ellitpic curve. If two such $E$ and $E'$ have the same height, then the associated ANSSes are the identical. The spectral sequences compute the homotopy groups of the spectrum $L_hS$, ($h=$ the height). There's a map $S\to L_hS$ from the sphere spectrum to this, so I guess we can say that $E$ detects the image of $\pi_*S\to \pi_* L_hS$.

Some details are in Hovey & Strickland, "Comodules and Landweber exact homology theories". (They really only show that the spectral sequences have the same $E_2$-term if the heights coincide.)

If you drop the condition that $E$ be Landweber exact, the precise nature of this will be a bit more complicated, but one still expects elliptic cohomology theories to "detect" only information up to height 2.

Here's an example to your explicit question. Let $C$ be any elliptic curve over the field $\mathbb{F}_p$. This will have a formal group of either height $h=1$ ("ordinary curve") or height $h=2$ ("supersingular curve"). For any such $C$, there is a ring spectrum $K_C$ with coefficient ring $\pi_* K_C=\mathbb{F}_p[u,u^{-1}]$, with $u\in \pi_2$, which is complex orientable, whose formal group is the formal group of $C$ ($K_C$ is the "2-periodic Morava $K$-theory" associated to the formal group).

Clearly, any such $K_C$ is an elliptic spectrum. The ANSS for $K_C$ captures height $h$ phenomena in the homotopy groups of spheres. Specifically, it computes the homotopy of the spectrum $L_{K(h)}S$, which is certainly different depending on whether $h$ is 1 or 2.

(If you want Landweber exact theories, you can use the theory $E_C$ associated to the "universal deformation" of the elliptic curve $C$, which will have coefficient ring $\pi_*E_C= \mathbb{Z}_p[[a]][u,u^{-1}]$ with $a\in\pi_0$ and $u\in \pi_2$.)

  • $\begingroup$ Let $E_1$ and $E_2$ be elliptic cohomology theories associated to elliptic curves of the same height over the same ring. If I understand correctly, you are saying that $E_1$ and $E_2$ have identical ANSS because (complex oriented, not sure if I need this here) cohomology theories of the same chromatic height have the same ANSS (i.e. elliptic cohomology theories of height $n$ have the same ANSS as $K(n)$). Do elliptic cohomology theories associated to elliptic curves of the same height also have identical Atiyah-Hirzeburch SS? $\endgroup$ Apr 1, 2015 at 4:37
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    $\begingroup$ That's approximately true. If two Landweber exact theories have the same heights, their associated AdamsNovikov s.s. (computing an approximation to the stable homotopy groups of spheres) are the same. This is called a "change of rings" theorem. The paper I cited explains some of this. (Without the Landweber exact hypothesis, it's not precisely true.) $\endgroup$ Apr 1, 2015 at 5:00

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