Duality for Topological Modular Forms

Details Duality for Topological Modular Forms Duality for Topological Modular Forms

2011-05-19

arxiv.org/abs/1105.3968v1

Mathematics

Algebraic Topology

31 pages, 6 figures, comments welcome

Scientific paper

It has been observed that certain localizations of the spectrum of topological modular forms $ tmf $ are self-dual (Mahowald-Rezk, Gross-Hopkins). We provide an integral explanation of these results that is internal to the geometry of the (compactified) moduli stack of elliptic curves $ \M $, yet is only true in the derived setting. When $ 2 $ is inverted, choice of level-$ 2 $-structure for an elliptic curve provides a geometrically well-behaved cover of $ \M $, which allows one to consider $ tmf $ as the homotopy fixed points of $ tmf(2) $, topological modular forms with level-$ 2 $-structure, under a natural action by $ GL_2(\Z/2) $. As a result of Grothendieck-Serre duality, we obtain that $ tmf(2) $ is self-dual. The vanishing of the associated Tate spectrum then makes $ tmf $ itself Anderson self-dual.

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