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Appendix III: Elliptic Curves and Topological

Modular Forms

In LN 1326 [1988] we have some of the first studies into the subject of elliptic co-
homology, and this subject was considered further by Hirzebruch, Berger, and Jung
[1994]. The first steps in the theory centered around the characteristic classes de-
scribed by series arising either from classical elliptic functions as in Chapters 9 and
10 or classical modular function as in Chapter 11.
In his 1994 ICM address Michael Hopkins proposed the cohomology theory or
spectrum, denoted tmf, of topological modular forms [1994]. Since that time, the
concept has gone through a development with still many of the basic results unpub-
lished. Now the manuscript for the 2002 ICM talk of Hopkins is available, we can see
a theory which brings new methods to old problems in homotopy theory including
the description of the homotopy groups of spheres.
Topological modular forms start with the Weierstrass equation and its related
change of variable as considered in Chapter 3. The coefficients in the Weierstrass
equation become indeterminates in a polynomial algebra, and this polynomial al-
gebra is extended with new indeterminants corresponding to the coefficients in the
change of variables. These polynomial algebras are linked by a structure called a
Hopf algebroid. With this Hopf algebroid we can return and give a description of the
category of elliptic curves and their isomorphisms.
Associated with an elliptic curve is a formal group law, see Chapter 12, §7. For-
mal groups also control the multiplicative properties of the first Chern class in gener-
alized cohomology theories, and for the complex bordism theory MU Quillen proved
that the formal group was the universal one parameter formal group, see Quillen
[1969]. This led to the possibility of making generalized cohomology theories from
a given formal group.
Hopf algebroids were first studied by J. F. Adams LN 99 [1969] pp. 1–138. Hopf
algebroids consist of two algebras with connecting data such that one algebra corre-
sponds to the coefficients of the theory and the other to the stable cooperations in the
2
theory. If there is an Adams spectral sequence in the theory, then the E -term should
be an Ext term over the operation algebra of the Hopf algebroid.
Hopf algebroids in a broader perspective are an example of a category object.
Categories as an organization of mathematical systems and their maps is a well es-