and as follows at the prime 3:
0 4 8 12 16 20 24 28 32 36 40 44 48
3 3
Here, a square indicates a copy of Z and a dot indicates a copy of Z/p. A little
number n drawn in a square indicates that the copy of Z in π∗(tmf ) maps onto an
index n subgroup of the corresponding Z in the ring of modular forms. A vertical
line between two dots indicates an additive extension, and a slanted line indicates
the multiplicative action of the generator η π1(tmf ) or ν π3(tmf ). The y-
coordinate, although vaguely reminiscent of the filtration degree in the Adams
spectral sequence, has no meaning in the above charts.
Note that, at the prime 2, the pattern on the top of the chart (that is, above
the expanding ko pattern on the base) repeats with a periodicity of 192 = 8 ·
24. A similar periodicity (not visible in the above chart) happens at the prime 3,
with period 72 = 3 · 24. Over Z, taking the least common multiple of these two
periodicities results in a periodicity of 24 · 24 = 576.
2. A brief history of tmf
In the sixties, Conner and Floyd proved that complex K-theory is determined
by complex cobordism: if X is a space, then its K-homology can be described as

= MU∗(X) ⊗MU∗ K∗, where K∗ is a module over the complex cobordism
ring of the point via the Todd genus map MU∗ K∗. Following this observation,
it was natural to look for other homology theories that could be obtained from
complex cobordism by a similar tensor product construction. By Quillen’s theorem
(1969), MU∗ is the base ring over which the universal formal group law is defined;
ring maps MU∗ R thus classify formal groups laws over R.
Given such a map, there is no guarantee in general that the functor X
MU∗(X) ⊗MU∗ R will be a homology theory. If R is a flat MU∗-module, then
long exact sequences remain exact after tensoring with R and so the functor in
question does indeed define a new homology theory. However, the condition of
being flat over MU∗ is quite restrictive. Landweber’s theorem (1976) showed that,
because arbitrary MU∗-modules do not occur as the MU-homology of spaces, the
flatness condition can be greatly relaxed. A more general condition, Landweber
exactness, suffices to ensure that the functor MU∗(−)⊗MU∗ R satisfies the axioms of
a homology theory. Shortly after the announcement of Landweber’s result, Morava
applied that theorem to the formal groups of certain elliptic curves and constructed
the first elliptic cohomology theories (though the term ‘elliptic cohomology’ was
coined only much later).
In the mid-eighties, Ochanine introduced certain genera (that is homomor-
phisms out of a bordism ring) related to elliptic integrals, and Witten constructed
a genus that took values in the ring of modular forms, provided the low-dimensional
characteristic classes of the manifold vanish. Landweber–Ravenel–Stong made ex-
plicit the connection between elliptic genera, modular forms, and elliptic coho-
mology by identifying the target of the universal Ochanine elliptic genus with the
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