Duality for topological modular forms
Summary: 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 \(\mathcal M\), 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 \(\mathcal M\), 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 \(\mathrm{GL}_2(\mathbb Z/2)\). 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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