Duality for topological modular forms

Abstract

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 $G{L}_2(\mathbb{Z}/2)$. 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.