A Survey of Elliptic Cohomology

AbstractHecke operators are used to investigate part of the E2-term of the Adams spectral sequence based on elliptic homology. The main result is a derivation of Ext1 which combines use of classical Hecke operators and p-adic Hecke operators due to Serre.

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The elliptic Weyl character formula

Compositio Mathematica ◽

10.1112/s0010437x1300777x ◽

2014 ◽

Vol 150 (7) ◽

pp. 1196-1234 ◽

Cited By ~ 9

Author(s):

Nora Ganter

Keyword(s):

Line Bundle ◽

Lie Groups ◽

Theoretical Framework ◽

Flag Variety ◽

Schubert Calculus ◽

Character Formula ◽

Elliptic Cohomology ◽

Image Position ◽

Weyl Character Formula

AbstractWe calculate equivariant elliptic cohomology of the partial flag variety$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}G/H$, where$H\subseteq G$are compact connected Lie groups of equal rank. We identify the${\rm RO}(G)$-graded coefficients${\mathcal{E}} ll_G^*$as powers of Looijenga’s line bundle and prove that transfer along the map$$\begin{equation*} \pi \,{:}\,G/H\longrightarrow {\rm pt} \end{equation*}$$is calculated by the Weyl–Kac character formula. Treating ordinary cohomology,$K$-theory and elliptic cohomology in parallel, this paper organizes the theoretical framework for the elliptic Schubert calculus of [N. Ganter and A. Ram,Elliptic Schubert calculus, in preparation].

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Elliptic cohomology of classifying spaces of cyclic groups and higher level modular forms

Journal of Mathematics of Kyoto University ◽

10.1215/kjm/1250518210 ◽

1997 ◽

Vol 37 (4) ◽

pp. 701-715

Author(s):

Shuma Morimoto ◽

Goro Nishida

Keyword(s):

Modular Forms ◽

Classifying Spaces ◽

Elliptic Cohomology ◽

Cyclic Groups

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An algebraic model for rational naïve-commutative ring SO(2)-spectra and equivariant elliptic cohomology

Mathematische Zeitschrift ◽

10.1007/s00209-020-02554-0 ◽

2020 ◽

Author(s):

David Barnes ◽

J. P. C. Greenlees ◽

Magdalena Kędziorek

Keyword(s):

Elliptic Curve ◽

Commutative Ring ◽

Equivariant Cohomology ◽

Algebraic Model ◽

Homotopy Groups ◽

Torsion Point ◽

Ring Spectrum ◽

Elliptic Cohomology ◽

Commutative Algebras ◽

Category Of Modules

Abstract Equipping a non-equivariant topological $$\text {E}_\infty $$ E ∞ -operad with the trivial G-action gives an operad in G-spaces. For a G-spectrum, being an algebra over this operad does not provide any multiplicative norm maps on homotopy groups. Algebras over this operad are called naïve-commutative ring G-spectra. In this paper we take $$G=SO(2)$$ G = S O ( 2 ) and we show that commutative algebras in the algebraic model for rational SO(2)-spectra model rational naïve-commutative ring SO(2)-spectra. In particular, this applies to show that the SO(2)-equivariant cohomology associated to an elliptic curve C of Greenlees (Topology 44(6):1213–1279, 2005) is represented by an $$\text {E}_\infty $$ E ∞ -ring spectrum. Moreover, the category of modules over that $$\text {E}_\infty $$ E ∞ -ring spectrum is equivalent to the derived category of sheaves over the elliptic curve C with the Zariski torsion point topology.

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