The Jacobi orientation and the two-variable elliptic genus
Matthew Ando, Christopher P French, Nora Ganter
ALGEBRAIC AND GEOMETRIC TOPOLOGY | GEOMETRY & TOPOLOGY PUBLICATIONS | Published : 2008
Let E be an elliptic spectrum with elliptic curve C. We show that the sigma orientation of Ando, Hopkins and Strickland [Invent. Math 146 (2001) 595687] and Hopkins [Proceedings of the ICM 12 (1995) 554565] gives rise to a genus of SU -manifolds taking its values in meromorphic functions on C. As C varies we find that the genus is a meromorphic arithmetic Jacobi form. When C is the Tate elliptic curve it specializes to the two variable elliptic genus studied by many. We also show that this two variable genus arises as an instance of the S1-equivariant sigma orientation. © 2008 Algebraic & Geometric Topology.