Elliptic cohomology and modular forms. (English) Zbl 0649.57022

Landweber, Peter S.

Elliptic curves and modular forms in algebraic topology, Proc. Conf., Princeton/NJ 1986, Lect. Notes Math. 1326 55-68 (1988).

[For the entire collection see Zbl 0642.00007.]
This paper describes how elliptic genera give rise to elliptic cohomology theory. These are cohomology theories which have coefficients given by various rings of modular forms. They are shown to exist by applying Quillen’s relations between complex bordism and formal groups to the classical addition formula for elliptic integrals of the first kind.
The paper also discusses integrality and divisibility results coming from elliptic genera and describes E. Witten’s formula for the universal elliptic genus.

Reviewer: R.E.Stong

MSC:

57R20	Characteristic classes and numbers in differential topology
11F11	Holomorphic modular forms of integral weight
55N35	Other homology theories in algebraic topology
14C40	Riemann-Roch theorems
14H15	Families, moduli of curves (analytic)

Keywords:

elliptic genera; elliptic cohomology theory; coefficients; rings of modular forms; complex bordism; formal groups; addition formula for elliptic integrals of the first kind; integrality; divisibility; universal elliptic genus