The Krichever Map and Automorphic Line Bundles

Abstract

A solution of the KP-hierarchy can be given by the τ -function or the Baker function associated to an element of the Grassmannian Gr(L 2(S 1)) consisting of some subspaces of the space L 2(S 1) of square-integrable functions on the unit circle S 1. The Krichever map associates an element W ∈ Gr(L 2(S 1)) to a line bundle over a Riemann surface equipped with some additional data. We consider a line bundle over a modular curve associated to an automorphy factor J and prove that the elements of the image W of this bundle under the Krichever map can be characterized by a set of criteria involving J.