Duality for topological modular forms
type: Item
instance of: scholarly article
title: Duality for topological modular forms
zbMATH Open document ID: 1366.55005
author: Vesna Stojanoska
published in: Documenta Mathematica
publication date: 2012-10-22T00:00:00Z
full work available at URL: https://arxiv.org/abs/1105.3968
full work available at URL: 10.html
review text: 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 \(\mathcal M\), 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 \(\mathrm{GL}_2(\mathbb Z/2)\). 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.
Mathematics Subject Classification ID: 55N34
Mathematics Subject Classification ID: 55N91
Mathematics Subject Classification ID: 55P43
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 14D23
zbMATH DE Number: 6097006
zbMATH Keywords: topological modular forms
zbMATH Keywords: Brown-Comenetz duality
zbMATH Keywords: generalized Tate cohomology
zbMATH Keywords: Serre duality
MaRDI profile type: Publication
arXiv ID: 1105.3968
description: scientific article