This is my attempt to understand Topological automorphic forms by Mark Behrens and Tyler Lawson. This 159 page memoir describes a general construction which includes that of the theory of topological modular forms (tmf) as a special case. It includes a lot of number theory and algebraic geometry not familiar to most topologists. Before attempting to read it one should look at Lawson's introduction, An overview of abelian varieties in homotopy theory, 37 pages. It describes some of the backgroup and motivation for the constructions and definitions in TAF. Another useful introduction is Topological algebraic geometry by Paul Goerss. This consists of over 100 slides for a series of lectures given in Copenhagen in June, 2008. Here you can find quick discussions of schemes, sheaves, stacks, topoi and other relevant notions from abstract algebraic geometry. For a record fast description of the moduli stack of elliptic curves over the complex numbers, see What is ... a stack? Goerss' writeup of his March, 2009 Bourbaki talk Topological Modular Forms [after Hopkins, Miller and Lurie], 33 pages, is the best single account I have seen of the construction of tmf. ArXiv link. For a very concise introduction to the relevant facts about elliptic curves and modular forms, see Appendix B (starting on page 53) of Ando-Hopkins-Strickland. Talks given in April, 2010, are based on Goerss' paper Quasi-coherent sheaves on the Moduli Stack of Formal Groups. |
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Created September 14, 2009; last modified August 29, 2011.