xxiv INTRODUCTION

maps X(n) → R correspond to coordinates up to degree n. Using this description,

one can show that MX(n) is the stack MFG,

(n)

the classifying stack of formal groups

with a coordinate up to degree n. The map from MFG

(n)

to MFG

(1)

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:

MX(4)∧tmf

∼

=

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

H∗(MSpin,

F2).

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

G3

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

C3.

That bundle is trivial, and because

C3

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

BO 8 → BO

J

− → 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

J

− → Σgl1(S) → Σgl1(tmf ).