#### Volume 8, issue 1 (2008)

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The Jacobi orientation and the two-variable elliptic genus

### Matthew Ando, Christopher P French and Nora Ganter

Algebraic & Geometric Topology 8 (2008) 493–539
##### 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.

##### Keywords
elliptic genus, jacobi forms, equivariant elliptic cohomology
Primary: 55N34
##### Publication
Received: 16 June 2006
Revised: 8 November 2007
Accepted: 13 November 2007
Published: 12 May 2008
##### Authors
 Matthew Ando Department of Mathematics The University of Illinois at Urbana-Champaign Urbana IL 61801 USA Christopher P French Department of Mathematics and Statistics Grinnell College Grinnell IA 50112 USA Nora Ganter Department of Mathematics Colby College Waterville ME 04901 USA