TITLE: What's an example of 2 elliptic curves with the same ground ring s.t. their associated cohomology theories detect different things?
QUESTION [14 upvotes]: 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):

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?

REPLY [9 votes]: 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 $(R_2)_*(X) \not\cong 0$.