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Topological modular and automorphic forms. (English) Zbl 1476.55018

Miller, Haynes (ed.), Handbook of homotopy theory. Boca Raton, FL: CRC Press. CRC Press/Chapman Hall Handb. Math. Ser., 221-261 (2020).
The Handbook of homotopy theory has been reviewed in [Zbl 1468.55001].
Summary: The spectrum of topological modular forms was first introduced by Hopkins and Miller. This chapter discusses a class of moduli stacks of abelian varieties which give rise to spectra of topological automorphic forms. Formal group laws also arise in the context of algebraic geometry. The chapter discusses two variants of the spectrum Tmf: the connective and the periodic versions. Integral versions of \(\mathcal{M}_{ell}\) can be defined by considering moduli spaces of elliptic curves with certain types of level structures. The most significant outstanding problem in the theory of topological modular forms is to give a geometric interpretation of this cohomology theory. The analysis of the 2-primary descent spectral sequence proceeds in a similar fashion, except that the computations are significantly more involved. Ando observed that power operations for elliptic cohomology are closely related to isogenies of elliptic curves.
For the entire collection see [Zbl 1468.55001].

MSC:

55N34 Elliptic cohomology
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
55P42 Stable homotopy theory, spectra
55-02 Research exposition (monographs, survey articles) pertaining to algebraic topology

Citations:

Zbl 1468.55001
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