xxviii INTRODUCTION

in turn isomorphic to the ´ etale cohomology of the stack Mell

ord

. In the end, the

essential calculation is

Hs(Mell ord

,

ω⊗k)

= 0 for s 0,

where k ∈ Z and ω is the line bundle of invariant differentials on Mell . At odd

primes, the obstruction groups vanish in the relevant degrees, thus proving the

existence and uniqueness of OK(1).

top

Unfortunately, the higher homotopy groups of

the space of lifts are not all zero, and so one doesn’t get a contractible space of

choices for the sheaf OK(1).

top

At the prime p = 2, one needs to use real instead of

complex K-theory to get obstruction groups that vanish.

The same obstruction theory for E∞-ring spectra also applies to E∞-ring maps,

such as the gluing maps αchrom and αarith for the Hasse square and the arith-

metic square. For αchrom, one considers the moduli space of all E∞-ring maps

ιord,∗OK(1)

top

→ ιss,∗(OK(2))K(1)

top

whose induced map of theta-algebras is prescribed,

and one tries to compute the homotopy groups of this moduli space. The obstruc-

tion groups here vanish, and there is an essentially unique map. For αarith, the

map is of rational spectra and the analysis is much easier; the obstruction groups

vanish, and again there is an essentially unique map.

The approach presented in Chapter 12 is a somewhat different way of con-

structing OK(1).

top

In this approach, one directly applies the K(1)-local obstruction

theory to construct LK(1)tmf, and then works backwards to construct

Otop.

That

approach allows one to avoid the obstruction theory for diagrams, but is more dif-

ficult in other steps—for instance, it requires use of level structures on the moduli

stack Mell to resolve the obstructions.

Chapter 13: The homotopy groups of tmf and of its localizations.

The homotopy groups of tmf are an elaborate amalgam of the classical ring of

modular forms MF∗ and certain pieces of the 2- and 3-primary part of the stable

homotopy groups of spheres π∗(S).

There are two homomorphisms

π∗(S) → π∗(tmf ) → MF∗.

The first map is the Hurewicz homomorphism, and it is an isomorphism on π0

through π6. Conjecturally, this map hits almost all of the interesting torsion classes

in π∗(tmf ) and its image (except for the classes η, η2, and ν) is periodic with period

576 (arising from a 192-fold periodicity at the prime 2 and a 72-fold periodicity

at the prime 3). Among others, the map is nontrivial on the 3-primary stable

homotopy classes α ∈ π3(S) and β ∈ π10(S) and the 2-primary stable homotopy

classes η, ν, , κ, κ, q ∈ π∗(S). The second map in the above display is the composite

of the inclusion π∗tmf → π∗Tmf with the boundary homomorphism in the elliptic

spectral sequence

Hs(Mell

;

πtOtop)

⇒ πt−s(Tmf ).

This map π∗(tmf ) → MF∗ = Z[c4,c6, Δ]/(c4

3

− c6

2

−

(12)3Δ)

is an isomorphism

after inverting the primes 2 and 3. The kernel of this map is exactly the torsion in

π∗(tmf ) and the cokernel is a cyclic group of order dividing 24 in degrees divisible

by 24, along with some number of cyclic groups of order 2 in degrees congruent to

4 mod 8. In particular, the map from π∗(tmf ) hits the modular forms c4, 2c6, and

24Δ, but c6 and Δ themselves are not in the image. The localization π∗(tmf )(p) at

any prime larger than 3 is isomorphic to (MF∗)(p)

∼

= Z(p)[c4,c6].