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

Abstract:
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

Topics:
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

• C: y² + y = x³

Lecture 3:
Formal groups from elliptic curves

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

Lecture 4:
Abstract formal group laws

Topics:
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 F_L over the universal ring L.

Lecture 5:
Formal group laws in characteristic p

Topics:
p-series; height; examples; the Honda formal group law F_n; the Morava stabilizer group S_n; complete local rings; the Witt ring WF_q; deformations of a formal group law over a complete local ring; isomorphisms of such.

Lecture 6:
Lubin-Tate deformation theory

Topics:
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 KU_p represents the universal deformation ring for height 1; J_p = L_K(1)S is the homotopy fixed points for the action by automorphisms of (F_p, F₁) on KU_p; the Adams summand L and the real K-theory spectrum KO.

Lecture 7:
Deformation of elliptic curves

Topics:
The case n=2; the lifted elliptic curve

• C': y² + a u x y + u³ y = x³

Lecture 8:
Generalized homology theories

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

[Lecture notes]

Lecture 9:
Complex bordism

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

[Lecture notes]

Lecture 10:
The Adams spectral sequence

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

Lecture 11:
The Steenrod algebra and its dual

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

Lecture 12:
The theorem of Milnor and Novikov

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

Lecture 13:
A theorem of Quillen

Topics:
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

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

Lecture 15:
The Morava homology theories E_n

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

Lecture 16:
The Hopkins-Miller theorem

Topics:
A_infty operads; A_infty ring spectra; the Hopkins-Miller theorem; the ring spectra E_n; action by the Morava stabilizer group S_n and the Galois automorphisms over F_p; the ring spectra EO_n for n = 1, 2.

[Lecture notes]

Lecture 17:
The Hopkins-Miller theory

Topics:
We outline the proof of the Hopkins-Miller theorem.

[Lecture notes]

Lecture 18:
Endomorphisms of formal group laws

Topics:
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 S_n.

Lecture 19:
The spectra EO₂ and eo₂

Topics:
The maximal finite group (of order 48) acting on E₂; outline of the Hopkins--Mahowald calculation of EO_2*, and their construction of eo₂.

Lecture 20:
Bockstein- and homotopy fixed point spectral sequences

Topics:
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 EO_2*.