Abstract:
In this introductory lecture we introduce elliptic curves, and the
so-called moduli space M_{1} of isomorphism classes of such.
The ring of polynomial functions defined on M_{1} is the ring of
integral modular forms, which was computed by Tate and described
by Deligne.
Then we introduce the Hopkins-Miller ring spectra E_{2} and EO_{2}, and the Hopkins-Mahowald connective cover eo_{2}, 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 eo_{2}, 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.
Topics:
Elliptic curves; the Riemann-Roch theorem; Weierstrass equations; the
group law; change of variables; the modular functions c_{4},
c_{6}, j and the discriminant; the supersingular elliptic curve
Topics:
More about the group G_{24}; 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.
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.
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.
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}, F_{1}) on KU_{p};
the Adams summand L and the real K-theory spectrum KO.
Topics:
The case n=2; the lifted elliptic curve
Topics:
Homology and cohomology theories; spectra; Brown's representability
theorem.
Topics:
Stably almost complex structures; complex bordism; the Thom spectrum
MU; the complex bordism ring.
Topics:
Exact couples, spectral sequences, convergence, the Adams spectral
sequence for a flat homology theory E_{*}.
Topics:
The Steenrod algebra and its dual; indecomposables and primitives;
the homology of MU.
Topics:
Milnor's calculation of the homotopy of MU;
the complex bordism ring.
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;
Topics:
Landweber's invariant prime ideal-, filtration- and exact functor
theorems.
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 E_{2};
a functor (FGLs) --> (complex oriented ring spectra).
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.
Topics:
We outline the proof of the Hopkins-Miller theorem.
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}.
Topics:
The maximal finite group (of order 48) acting on E_{2}; outline
of the Hopkins--Mahowald calculation of EO_{2*}, and their
construction of eo_{2}.
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.
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).
Kozul resolutions; the Kozul spectral sequence; Ext over A(1); the Adams spectral sequence for ko.
Ext over A(2); the Adams E_{2} term for eo_{2}.
A complex with cohomology the `double' of A(1).
The Bockstein and Adams-Novikov spectral sequences for EO_{2*}.