The Jacobi orientation and the two-variable elliptic genus

Ando, Matthew¹ ; French, Christopher P² ; Ganter, Nora³

¹Department of Mathematics, The University of Illinois at Urbana-Champaign, Urbana IL 61801, USA
²Department of Mathematics and Statistics, Grinnell College, Grinnell IA 50112, USA
³Department of Mathematics, Colby College, Waterville ME 04901, USA

Algebraic and Geometric Topology, Tome 8 (2008) no. 1, pp. 493-539

Résumé

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.

Détail

DOI : 10.2140/agt.2008.8.493

Keywords: elliptic genus, jacobi forms, equivariant elliptic cohomology

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