  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.
