maps X(n) R correspond to coordinates up to degree n. Using this description,
one can show that MX(n) is the stack MFG,
the classifying stack of formal groups
with a coordinate up to degree n. The map from MFG
to MFG
is the obvious
forgetful map.
The stack Mtmf associated to tmf is the moduli stack of generalized elliptic
curves (both multiplicative and additive degenerations allowed) with first order
coordinate. The stack MX(4)∧tmf can therefore be identified with the moduli stack
of elliptic curves together with a coordinate up to degree 4. The pair of an elliptic
curve and such a coordinate identifies a Weierstrass equation for the curve, and so
this stack is in fact a scheme:

Spec Z[a1,a2,a3,a4,a6].
Here, a1,a2,a3,a4,a6 are the coefficients of the universal Weierstrass equation. By
considering the products X(4) . . . X(4) tmf , one can furthermore identify
the whole X(4)-based Adams resolution of tmf with the cobar resolution of the
Weierstrass Hopf algebroid.
Chapter 10: The string orientation. The string orientation, or σ-orienta-
tion of tmf is a map of E∞-ring spectra
MO 8 tmf .
Here, MO 8 = MString is the Thom spectrum of the 7-connected cover of BO, and
its homotopy groups are the cobordism groups of string manifolds (manifolds with
a chosen lift to BO 8 of their tangent bundle’s classifying map). At the level of
homotopy groups, the map MO 8 tmf is the Witten genus, a homomorphism
[M] φW (M) from the string cobordism ring to the ring of integral modular
forms. Note that φW (M) being an element of π∗(tmf ) instead of a mere modular
form provides interesting congruences, not visible from the original definition of the
Witten genus.
Even before having a proof, there are hints that the σ-orientation should exist.
The Steenrod algebra module H∗(tmf , F2) = A/ /A(2) occurs as a summand in
H∗(MString, F2). This is reminiscent of the situation with the Atiyah–Bott–Shapiro
orientation MSpin ko, where H∗(ko, F2) = A/ /A(1) occurs as a summand of
Another hint is that, for any complex oriented cohomology theory E with
associated formal group G, multiplicative (not E∞) maps MO 8 E correspond
to sections of a line bundle over
subject to a certain cocycle condition. If G is the
completion of an elliptic curve C, then that line bundle is naturally the restriction
of a bundle over
That bundle is trivial, and because
is proper, its space of
sections is one dimensional (and there is even a preferred section). Thus, there is a
preferred map MO 8 E for every elliptic spectrum E.
A not-necessarily E∞ orientation MO 8 tmf is the same thing as a nullho-
motopy of the composite
BGL1(S) BGL1(tmf ),
where BGL1(R) is the classifying space for rank one R-modules. An E∞ orientation
MO 8 tmf is a nullhomotopy of the corresponding map of spectra
bo 8 bo
Σgl1(S) Σgl1(tmf ).
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