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426 Appendix III: Elliptic Curves and Topological Modular Forms

tablished perspective in mathematics. The mathematicians differ on to what extent
the categorical framework should be made explicit, but almost everybody agrees that
it can be a useful way to look at certain phenomena.
Since Mumford’s study of Pic on the moduli of elliptic curves [1968], Quillen’s
work on algebraic K-theory [1972], and Adams’ work on generalized cohomology
theories [1969], people have studied categories as one studies groups, topological
spaces, or Hopf algebras. This led to such ideas as the category of small categories,
or more generally, the category Cat(C) of category objects in a category C.Aspe-
cial case are groupoids, they are categories where all morphisms are isomorphisms.
This means that there is another concept of Grpoid(C) which denotes the category of
groupoids in the category C.
In the work of Mumford and further the work of Deligne and Mumford groupoids
were especially important. In the work of Adams we have the opposite concept of
a cocategory in the category of commutative algebras which is also called a Hopf
algebroid. Many of the considerations coming into the work of Mumford and Quillen
were already anticipated by Grothendieck.
The aim of this appendix is to reexamine the Weierstrass polynomial and its
change of variables introduced in Chapter 3, and to put the data into a Hopf algebroid.
This example becomes an algebraic motivation for the definition of Hopf algebroid.
We show how to describe the category of elliptic curves and isomorphisms and how
to determine the ring of modular forms in terms of the Weierstrass Hopf algebroid.
This Hopf algebroid plays a basic role in describing the new cohomology theory
tmf called topological modular forms. There is a spectral sequence with E 0,∗ the
2
ring of modular forms, and it converges to π ∗ (tmf). Under the edge morphism to
E 0,∗ the torsion in π ∗ (tmf) goes to zero and certain modular forms like are not in
2
the image, but 24 is in the image. The edge morphism is a rational isomorphism.
Since the polynomials in coefficients of the Weierstrass equation generate the ring of
modular forms, we see the motivation for the term topological modular forms as a
name for the cohomology theory.
With this discussion the reader has some background for the study of all of these
new developments in homotopy theory related to topological modular forms.
This appendix is an elementary introduction to the bridge between elliptic curves
as defined in Chapter 3 by Weierstrass equations with changes of variable and the
Hopf algebroid picture used in studying cohomology theories. We begin with a dis-
cussion of categories and groupoids in a cateory and then consider the concept of
Hopf algebroid which is a cogroupoid in the category of commutative algebras. This
elementary material is included so to fix notation and the basic definitions. Then the
Weierstrass Hopf algebroid is introduced, and its relation to isomorphism classes of
elliptic curves is considered. The Weierstrass Hopf algebroid is used to compute the
homotopy of the spectrum topological modular forms, denoted tmf. The construc-
tions of tmf is still work in progress at this time.
Apart from a general sketch of ideas and suitable references to the literature, any
real development of this theory would be beyond the scope of this appendix.