Mike Hopkins and Mark Mahowald, From Elliptic Curves to Homotopy Theory,
preprint, MIT and Northwestern University, June 1998 (here is the PostScript file).
Weeks 4 and 5
For basic material on elliptic curves, read:
Joseph H. Silverman, The Arithmetic of Elliptic Curves,
Graduate Texts in Mathematics no. 106, Springer-Verlag, 1986
(available at the Akademika bookstore).
- Chapter II: Algebraic Curves,
II.1 Curves, II.5 The Riemann-Roch Theorem.
- Chapter III: The Geometry of Elliptic Curves,
III.1 Weierstrass Equations, III.2 The Group Law, III.3 Elliptic Curves,
III.10 The Automorphism Group.
- Chapter IV: The Formal Group of an Elliptic Curve,
IV.1 Expansion around O, IV.2 Formal Groups, IV.3 Groups Associated to
Formal Groups, IV.4 The Invariant Differential, IV.7 Formal Groups in
- Appendix C.
Section 12 Modular Functions.
Weeks 6 and 7
For material on formal group laws, read one of:
John Frank Adams, Stable Homotopy and Generalised Homology,
Chicago Lectures in Mathematics, The University of Chicago Press, 1974
(available at amazon.com.)
- Part II: Quillen's work on formal groups and complex cobordism.
Douglas C. Ravenel, Complex Cobordism and Stable Homotopy Groups of Spheres
Pure and Applied Mathematics, Academic Press, Inc., 1986
(out of print, but you may try amazon.com.)
- Appendix 2: Formal Group Laws.
or Frölich or Hazewinkel's books.
For the Lubin-Tate deformation theory, I will use the presentation
in Charles Rezk's Notes on the
Week 8 (February 22nd - 26th) I'll be in Bielefeld for the workshop.
Weeks 9 and 10
For the Steenrod algebra, exact couples,
the Adams spectral sequence, and the complex bordism ring
the following may be useful:
John Rognes / January 26th 1999
Norman E. Steenrod and D.B.A. Epstein, Cohomology operations,
Mathematics Studies no. 50, Princeton University Press, 1962.
Robert E. Mosher and Martin C. Tangora,
Cohomology operations and applications in homotopy theory,
Harper's Series in Modern Mathematics, Harper and Row, 1968.
J. Michael Boardman, Conditionally Convergent Spectral Sequences,
preprint, The Johns Hopkins University, 1981 (paper copy available from
John W. Milnor,
The Steenrod algebra and its dual,
Ann. of Math., vol. 67, pages 150-171 (1958).
John W. Milnor,
On the cobordism ring $\Omega^*$ and a complex analogue. I,
Amer. J. Math., vol. 82, pages 505-521 (1960).