Duality for Topological Modular Forms

It has been observed that certain localizations of the spectrum of topological modular forms 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, a 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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