Injectivity of the genus 1 Kudla-Millson lift on locally symmetric spaces

Abstract

Let $L$ be an even indefinite lattice. We show that if $L$ splits off a hyperbolic plane and a scaled hyperbolic plane, then the Kudla-Millson lift of genus $1$ associated to $L$ is injective. Our result includes as special cases all previously known injectivity results on the whole space of elliptic cusp forms available in the literature. In particular, we also consider the Funke-Millson twist of the lift. Further, we provide geometric applications on locally symmetric spaces of orthogonal type.