Geodesic
Duality for topological modular forms
Documenta mathematica, Tome 17 (2012), pp. 271-311.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

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.
Classification : 55N34, 55N91, 55P43, 14H52, 14D23
Keywords: topological modular forms, Brown-comenetz duality, generalized Tate cohomology, Serre duality 
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     author = {Stojanoska, Vesna},
     title = {Duality for topological modular forms},
     journal = {Documenta mathematica},
     pages = {271--311},
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     volume = {17},
     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2012__17__a20/}
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Stojanoska, Vesna. Duality for topological modular forms. Documenta mathematica, Tome 17 (2012), pp. 271-311. http://geodesic.mathdoc.fr/item/DOCMA_2012__17__a20/