xvi INTRODUCTION
coefficient ring of the homology theory X MU∗(X) ⊗MU∗ Z[
1
2
][δ, ,
Δ−1]
asso-
ciated to the Jacobi quartic elliptic curve y2 = 1 2δx2 + x4 (here, Δ is the
discriminant of the polynomial in x). Segal had also presented a picture of the re-
lationship between elliptic cohomology and Witten’s physics-inspired index theory
on loop spaces. In hindsight, a natural question would have been whether there
existed a form of elliptic cohomology that received Witten’s genus, thus explaining
its integrality and modularity properties. But at the time, the community’s at-
tention was on Witten’s rigidity conjecture for elliptic genera (established by Bott
and Taubes), and on finding a geometric interpretation for elliptic cohomology—a
problem that remains open to this day, despite a tantalizing proposal by Segal and
much subsequent work.
Around 1989, inspired in part by work of McClure and Baker on A∞ structures
and actions on spectra and by Ravenel’s work on the odd primary Arf invariant,
Hopkins and Miller showed that a certain profinite group known as the Morava
stabilizer group acts by A∞ automorphisms on the Lubin–Tate spectrum En (the
representing spectrum for the Landweber exact homology theory associated to the
universal deformation of a height n formal group law). Of special interest was
the action of the binary tetrahedral group on the spectrum E2 at the prime 2.
The homotopy fixed point spectrum of this action was called EO2, by analogy
with the real K-theory spectrum KO being the homotopy fixed points of complex
conjugation on the complex K-theory spectrum.
Mahowald recognized the homotopy of EO2 as a periodic version of a hypo-
thetical spectrum with mod 2 cohomology A/ /A(2), the quotient of the Steenrod
algebra by the submodule generated by Sq1, Sq2, and Sq4. It seemed likely that
there would be a corresponding connective spectrum eo2 and indeed a bit later
Hopkins and Mahowald produced such a spectrum; (in hindsight, that spectrum
eo2 is seen as the 2-localization of tmf ). However, Davis–Mahowald (1982) had
proved, by an intricate spectral sequence argument, that it is impossible to real-
ize A/ /A(2) as the cohomology of a spectrum. This conundrum was resolved only
much later, when Mahowald found a missing differential around the
55th
stem of the
Adams spectral sequence for the sphere, invalidating the earlier Davis–Mahowald
argument.
In the meantime, computations of the cohomology of MO 8 at the prime 2
revealed an A/ /A(2) summand, suggesting the existence of a map of spectra from
MO 8 to eo2. While attempting to construct a map MO 8 EO2, Hopkins
(1994) thought to view the binary tetrahedral group as the automorphism group of
the supersingular elliptic curve at the prime 2; the idea of a sheaf of ring spectra
over the moduli stack of elliptic curves quickly followed—the global sections of that
sheaf, TMF , would then be an integral version of EO2.
The language of stacks, initially brought to bear on complex cobordism and
formal groups by Strickland, proved crucial for even formulating the question TMF
would answer. In particular, the stacky perspective allowed a reformulation of
Landweber’s exactness criterion in a more conceptual and geometric way: MU∗
R is Landweber exact if and only if the corresponding map to the moduli stack of
formal groups, Spec(R) MFG , is flat. From this viewpoint, Landweber’s theorem
defined a presheaf of homology theories on the flat site of the moduli stack MFG of
formal groups. Restricting to those formal groups coming from elliptic curves then
provided a presheaf of homology theories on the moduli stack of elliptic curves.
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