MA 422, Spring 1999
Elliptic curves and chromatic homotopy theory
John Rognes

Lecture 1:
Introduction to elliptic curves and the stable homotopy groups of spheres

In this introductory lecture we introduce elliptic curves, and the so-called moduli space M1 of isomorphism classes of such. The ring of polynomial functions defined on M1 is the ring of integral modular forms, which was computed by Tate and described by Deligne.

Then we introduce the Hopkins-Miller ring spectra E2 and EO2, and the Hopkins-Mahowald connective cover eo2, whose homotopy groups are the topological modular forms. There is a natural map from topological modular forms to modular forms, which becomes an equivalence upon inverting 2 and 3.

Next we take a look at the stable homotopy groups of spheres, which is the ground ring for stable homotopy theory, somewhat like the integers Z is the ground ring for algebra. The calculation of these groups has proceeded somewhat like the evolution of natural sciences, and is progressing from the empirical to the theoretical stage. We now know that the stable homotopy groups of spheres arise in periodic families, and thus admit a chromatic decomposition by their periods, in a manner similar to how light waves have different colors, determined by their wave-lengths.

At the high-energy end of the chromatic spectrum we find the chromatic type 0 elements which are detected by rational homology. This is the 0-dimensional homotopy group, where the stable algebraic topology is indistinguishable from algebra. Next come the chromatic type 1 elements which are detected by topological K-theory. This is the so-called image of J in the stable homotopy groups of spheres, which appears periodically every 8 dimensions.

The subject of the present course is the next family of elements in the stable homotopy groups of spheres, namely the chromatic type 2 elements. These are detected by the topological modular forms theory eo2, which was constructed using elliptic curves, and appear periodically every 192 dimensions.

It is the aim of this course to follow the work of Hopkins and Mahowald, leading to the identification of these chromatic type 2 elements in the stable homotopy groups of spheres, and to learn a bit about elliptic curves and stable algebraic topology along the way. This first lecture will be introductory, will prove nothing, and will presume a minimum of mathematical background.

Lecture 2:
Elliptic curves

Elliptic curves; the Riemann-Roch theorem; Weierstrass equations; the group law; change of variables; the modular functions c4, c6, j and the discriminant; the supersingular elliptic curve

over F4; its group law; its binary tetrahedral automorphism group G24.

Lecture 3:
Formal groups from elliptic curves

More about the group G24; expansion around the origin O; formal groups; additive, multiplicative and Euler elliptic formal group laws; the formal group FC of an elliptic curve C.

Lecture 4:
Abstract formal group laws

Survey of the path ahead; formal group laws associated to singular cohomology, complex K-theory; the category FGL(R) of formal group laws over R; isomorphisms; strict isomorphisms; change of rings; Lazard's universal formal group law FL over the universal ring L.

Lecture 5:
Formal group laws in characteristic p

p-series; height; examples; the Honda formal group law Fn; the Morava stabilizer group Sn; complete local rings; the Witt ring WFq; deformations of a formal group law over a complete local ring; isomorphisms of such.

Lecture 6:
Lubin-Tate deformation theory

Universal deformations; the Lubin-Tate theorem; the Lubin-Tate ring; functoriality and group actions; extensions to degree -2 formal group laws; the case n=1; p-adic complex K-theory KUp represents the universal deformation ring for height 1; Jp = LK(1)S is the homotopy fixed points for the action by automorphisms of (Fp, F1) on KUp; the Adams summand L and the real K-theory spectrum KO.

Lecture 7:
Deformation of elliptic curves

The case n=2; the lifted elliptic curve

over the extended universal ring WF4[[a]][u,u-1]; its formal group law; the action of G24 on WF4[[a]][u,u-1].

Lecture 8:
Generalized homology theories

Homology and cohomology theories; spectra; Brown's representability theorem.

[Lecture notes]

Lecture 9:
Complex bordism

Stably almost complex structures; complex bordism; the Thom spectrum MU; the complex bordism ring.

A look ahead:
Theorems of Milnor and Quillen; Landweber's exact functor theorem; the Hopkins-Miller ring spectra En.

[Lecture notes]

Lecture 10:
The Adams spectral sequence

Exact couples, spectral sequences, convergence, the Adams spectral sequence for a flat homology theory E*.

Lecture 11:
The Steenrod algebra and its dual

The Steenrod algebra and its dual; indecomposables and primitives; the homology of MU.

Lecture 12:
The theorem of Milnor and Novikov

Milnor's calculation of the homotopy of MU; the complex bordism ring.

Lecture 13:
A theorem of Quillen

Complex oriented theories; the Atiyah-Hirzebruch spectral sequence; first calculations; the associated formal group law; a functor (complex oriented ring spectra) --> (FGLs); the universal formal group law over Lazard's ring L; Quillen's theorem; calculations in E-homology and cohomology; ring spectrum maps MU --> E;

Lecture 14:
Landweber's exact functor theorem

Landweber's invariant prime ideal-, filtration- and exact functor theorems.

Lecture 15:
The Morava homology theories En

The Araki generators vn; the Brown-Peterson spectrum BP; the Johnson-Wilson spectra E(n); the Morava homology theories Ek,G and En; proof of Landweber exactness for E2; a functor (FGLs) --> (complex oriented ring spectra).

Lecture 16:
The Hopkins-Miller theorem

Ainfty operads; Ainfty ring spectra; the Hopkins-Miller theorem; the ring spectra En; action by the Morava stabilizer group Sn and the Galois automorphisms over Fp; the ring spectra EOn for n = 1, 2.

[Lecture notes]

Lecture 17:
The Hopkins-Miller theory

We outline the proof of the Hopkins-Miller theorem.

[Lecture notes]

Lecture 18:
Endomorphisms of formal group laws

The endomorphism ring of a formal group law of finite height over a field of positive characteristic, its group of units, cohomological properties of the Morava stabilizer group Sn.

Lecture 19:
The spectra EO2 and eo2

The maximal finite group (of order 48) acting on E2; outline of the Hopkins--Mahowald calculation of EO2*, and their construction of eo2.

Lecture 20:
Bockstein- and homotopy fixed point spectral sequences

Homotopy of KU, E(n), ku, BP<n>; the Bockstein spectral sequence for KO, ko; the homotopy fixed point spectral sequence for KO, ko; subalgebras E(n), A(n), E of the Steenrod algebra A.

Lecture 21:
Ext over subalgebras of the Steenrod algebra

Cohomology of ku, BP<n>, BP; the Adams spectral sequence for ku, BP<n>, BP; cohomology of ko, bo, bso, bspin; a resolution for A(1).

Lecture 22:
The Kozul spectral sequence

Kozul resolutions; the Kozul spectral sequence; Ext over A(1); the Adams spectral sequence for ko.

Lecture 23:
The Adams spectral sequence for eo2

Ext over A(2); the Adams E2 term for eo2.

Lecture 24:
The complex DA1

A complex with cohomology the `double' of A(1).

Lecture 25:
On the homotopy of EO2

The Bockstein and Adams-Novikov spectral sequences for EO2*.