Duality for Topological Modular Forms

TL;DR

This paper provides an internal geometric explanation for the self-duality properties of topological modular forms spectra, using derived algebraic geometry and level structures on elliptic curves.

Contribution

It offers a new integral geometric perspective on the duality phenomena in topological modular forms, connecting derived geometry with classical duality results.

Findings

$Tmf(2)$ is self-dual due to Grothendieck-Serre duality.

The Tate spectrum vanishes, leading to $Tmf$ being Anderson self-dual.

Provides an integral, geometric explanation for known duality results.

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 $ \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.