Duality for topological modular forms
By Vesna Stojanoska. Documenta Mathematica, 2011
58 citations recorded by the source corpus.
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 GL2(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.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/4a4be5486e3ad6134ed05b33f8c6f83393b194c2
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Citations in the corpus
- Topological modular forms with level structure (2013, background): “There aremany situationswhere extra functoriality for elliptic spectra can be a great advantage [7,37,51].”
- The C2–spectrum Tmf1(3) and its invertible modules (2015, methodology): “For example, Stojanoska computed in [60] that Tmf 1 2 is Anderson self-dual in the sense that IZŒ 1 2 Tmf 1 2 '†(21) Tmf 1 2 .”
- Affineness and chromatic homotopy theory (2013, background): “Note furthermore that the second theorem was proved by Stojanoska in [50] in the case n = 2 in her investigation of the Anderson selfduality of Tmf.”
- On Hopkins’ Picard groups for the prime 3 and chromatic level 2 (2012, background): “The equivariantly minded reader will recognize β−1 1 E hG24 2 as the Tate spectrum of the G24-action on E2; see [9, 26].”