On the Injectivity of the Kudla-Millson Lift and Surjectivity of the Borcherds Lift

Summary

Abstract

We consider the Kudla-Millson lift from elliptic modular forms of weight (p + q)/2 to closed q-forms on locally symmetric spaces corresponding to the orthogonal group O(p, q). We study the L²-norm of the lift following the Rallis inner product formula. We compute the contribution at the Archimedian place. For locally symmetric spaces associated to even unimodular lattices, we obtain an explicit formula for the L²-norm of the lift, which often implies that the lift is injective. For O(p, 2) we discuss how such injectivity results imply the surjectivity of the Borcherds lift.

Introduction

In previous work, we studied the Kudla-Millson theta lift (see e.g.) and Borcherds' singular theta lift (e.g.) and established a duality statement between these two lifts. Both of these lifts have played a significant role in the study of certain cycles in locally symmetric spaces and Shimura varieties of orthogonal type. In this paper, we study the injectivity of the Kudla-Millson theta lift, and revisit part of the material of from the viewpoint of, to obtain surjectivity results for the Borcherds lift. Moreover, we provide evidence for the following principle: The vanishing of the standard L-function of a cusp form of weight 1 + p/2 at s₀ = p/2 corresponds to the existence of a certain “exceptional automorphic product” on O(p, 2) (see Theorem 1.8).