Elliptic Cohomology and Elliptic Curves (felix Klein

Lecture notes for a series of talks given in Bonn, June 2015. Most of the topics covered touched in one way or another on the role of power operations in elliptic cohomology. In June of 2015 I gave a series of six lectures (the Felix Klein lectures) in Bonn. These are some of my lecture notes for those talks. I had hoped to polish them more carefully, but that hasn’t happend yet, and at this point probably will not. I am making them available more or less as-is. I include only the notes for the first five lectures. Note that some bits in these notes never made it into the spoken lectures. The notes for the final lecture are too disjointed to be very useful, so I have omitted them. I hope to soon have preprints on some aspects of what I spoke about in that lecture. I’d like to thank the Hausdorff Institute for their hospitality, and for the opportunity to give these talks, which were a great challenge, but also great fun. 1. What is elliptic cohomology? I’ll start with a brief ”pseudo-historical” account of elliptic cohomology. This is meant to be an imprecise overview. The idea is to introduce the basic questions and objects we’re interested in, and to highlight the main themes of these lectures, which could be summarized as “power operations” and “isogenies”. 1.1. Genera. A genus is a function which assigns to each closed manifold M of some type an element Φ(M) ∈ R of a commutative ring R, satisfying • Φ(M1 qM2) = Φ(M1) + Φ(M2). • Φ(M1 ×M2) = Φ(M1)Φ(M2). • Φ(∂N) = 0. This is the same as giving a ring homomorphism from a suitable cobordism ring, e.g., Φ: MSO∗ → R or Φ: MU∗ → R. Genera with values in R with Q ⊂ R can be described entirely in terms of characteristic classes, by a formalism due to Hirzebruch. For instance, associated to a genus Φ: MU∗ → R⊗Q is a characteristic class for complex vector bundles KΦ(V → X) ∈ H∗(X;R⊗Q), which is completly determined by its characteristic series, i.e., its value on the universal line bundle KΦ(x) = KΦ(O(1)→ BU(1)) ∈ H∗(BU(1);R⊗Q) = R⊗Q[[x]] Date: June 6, 2016. This work was partially supported by the National Science Foundation, DMS-1406121. 1See for instance, [HBJ92].