Volume 8, issue 1 (2008)

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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
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 S1–equivariant sigma orientation.

Keywords
elliptic genus, jacobi forms, equivariant elliptic cohomology
Mathematical Subject Classification 2000
Primary: 55N34
References
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