By Christopher L. Douglas, John Francis, André G. Henriques, Michael A. Hill, Andre G. Henriques

The speculation of topological modular varieties is an problematic combination of classical algebraic modular types and strong homotopy teams of spheres. the development of this idea combines an algebro-geometric point of view on elliptic curves over finite fields with suggestions from algebraic topology, really strong homotopy idea. It has functions to and connections with manifold topology, quantity conception, and string thought. This e-book offers a cautious, obtainable advent to topological modular types. After a short historical past and a longer assessment of the topic, the ebook right commences with an exposition of classical facets of elliptic cohomology, together with historical past fabric on elliptic curves and modular types, an outline of the moduli stack of elliptic curves, an evidence of the precise functor theorem for developing cohomology theories, and an exploration of sheaves in good homotopy idea. There follows a remedy of extra really good subject matters, together with localization of spectra, the deformation idea of formal teams, and Goerss-Hopkins obstruction idea for multiplicative constructions on spectra. The publication then proceeds to extra complex fabric, together with discussions of the string orientation, the sheaf of spectra at the moduli stack of elliptic curves, the homotopy of topological modular kinds, and an in depth account of the development of the spectrum of topological modular kinds. The booklet concludes with the 3 unique, pioneering and vastly influential manuscripts at the topic, through Hopkins, Miller, and Mahowald

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2) Take the line L passing through R and e = [0, 1, 0] and deﬁne P + Q to be the third point of intersection in L ∩ C. Here is a picture that illustrates this group law: R Q L P P+Q L′ Group law on an elliptic curve. 1. The law deﬁned above provides an abelian group structure on the points of C with identity element e = [0, 1, 0]. In fact, the maps + : E × E → E, − : E → E are morphisms. 2-3. 2. An elliptic curve C over K is a one dimensional group variety over K. In fact, there is a rather strong converse to this statement.

1005, viii, 221–255, 2010. S´ eminaire Bourbaki. Volume 2008/2009. Expos´es 997–1011. MR2648680 (2011m:55003) [Hop] M. J. Hopkins. Algebraic topology and modular forms. In Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), pages 291–317, Beijing, 2002. Higher Ed. Press. MR1989190 (2004g:11032) [Lan] P. S. Landweber, editor. Elliptic curves and modular forms in algebraic topology, volume 1326 of Lecture Notes in Mathematics, Berlin, 1988. Springer-Verlag. MR970278 (91a:57021) [Lur] Jacob Lurie.

The functors ΩG ∗ are examples of generalized homology theories, and the Pontryagin–Thom construction shows they are represented by the Thom spectra M G = {M Gk } = {T h(ρ∗k ξk )}. Here, ξk → BO(k) is the universal k-dimensional vector bundle (ξk = EO(k) ×O(k) Rk ), and for any vector bundle V → X the Thom space T h(V ) is deﬁned as the unit disc bundle modulo the unit sphere bundle D(V )/S(V ). Particularly common examples of G-bordism include oriented bordism, spin bordism, and complex bordism, corresponding to the groups SO(k), Spin(k), and U (k), respectively.