Let α and β be any angles then the known formula sin (α+β) = sinα cosβ + cosα sinβ becomes under the substitution x = sinα, y = sinβ, sin (α + β) = x √(1 - y2) + y √(1 - x2) =: F(x,y). This addition formula is an example of "Formal group law", which show up in many contexts in Modern Mathematics.
In algebraic topology suitable cohomology theories induce a Formal group Law, the elliptic cohomologies are the ones who realize the Euler addition formula (1778): F(x,y) =: (x √R(y) + y √R(x)/1 - εx2y2). For R(z) = 1 - 2δz2 + εz4 the above case corresponds to ε=0, δ=1/2.
In this survey paper we define these cohomology theories and establish their relationship with global analysis (Atiyah-Singer theorem) and modular forms following ideas of Landweber, Hirzebruch et al.
@article{urn:eudml:doc:41753,
title = {Elliptic cohomologies: an introductory survey.},
journal = {Publicacions Matem\`atiques},
volume = {36},
year = {1992},
pages = {789-806},
mrnumber = {MR1210020},
zbl = {0777.55005},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41753}
}
Moreno, Guillermo. Elliptic cohomologies: an introductory survey.. Publicacions Matemàtiques, Tome 36 (1992) pp. 789-806. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41753/