Duality for topological modular forms

<jats:p> 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 <jats:inline-formula> <jats:tex-math>\mathcal M</jats:tex-math> </jats:inline-formula> , 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 <jats:inline-formula> <jats:tex-math>\mathcal M</jats:tex-math> </jats:inline-formula> , which allows one to consider <jats:inline-formula> <jats:tex-math>Tmf</jats:tex-math> </jats:inline-formula> as the homotopy fixed points of <jats:inline-formula> <jats:tex-math>Tmf(2)</jats:tex-math> </jats:inline-formula> , topological modular forms with level 2 structure, under a natural action by <jats:inline-formula> <jats:tex-math>GL_2(\mathbb Z/2)</jats:tex-math> </jats:inline-formula> . As a result of Grothendieck–Serre duality, we obtain that <jats:inline-formula> <jats:tex-math>Tmf(2)</jats:tex-math> </jats:inline-formula> is self-dual. The vanishing of the associated Tate spectrum then makes <jats:inline-formula> <jats:tex-math>Tmf</jats:tex-math> </jats:inline-formula> itself Anderson self-dual. </jats:p>