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.