## Duality for topological modular forms

### Summary

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 $\M$, 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.

### Mathematics Subject Classification

55N34, 55N91, 55P43, 14H52, 14D23

### Keywords/Phrases

topological modular forms, Brown-comenetz duality, generalized Tate cohomology, Serre duality