 
Summary: ELLIPTIC CURVES AND ALGEBRAIC TOPOLOGY
MATTHEW ANDO
Part 1. Elliptic curves and chromatic stable homotopy theory
Elliptic curves enter algebraic topology through "Elliptic cohomology"really a family of cohomology
theoriesand their associated "elliptic genera".
· Arithmetic aspect: Modularity of elliptic genera, The spectrum TMF of "topological modular forms"
and the calculation of TMF MF(Z), Hopkins's proof of Borcherds' congruences.
· Physical aspect: Witten's approach to elliptic genera via string theory.
· Homotopy theoretic aspect: Relationship to chromatic program, Hopkins and Mahowald's calculation
of S TMF.
1. The Adams spectral sequence and descent
Forgetful functor
Not quite faithful. Instead
Adams Spectral Sequence
Let E be a generalized (co)homology theory. One could hope to study maps from X to Y by studying
maps
EX EY
(of EEcomodules).
For example, consider
S1 n
