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].
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