The Jacobi orientation and the two-variable elliptic genus

Abstract

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) 595-687] and Hopkins [Proceedings of the ICM 1-2 (1995) 554-565] 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 $S^1$–equivariant sigma orientation.