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

TL;DR

This paper proves the injectivity of the genus 1 Kudla-Millson lift for certain lattices, extending previous results and providing geometric applications on locally symmetric spaces of orthogonal type.

Contribution

It establishes new injectivity conditions for the Kudla-Millson lift when the lattice splits off hyperbolic planes, including the Funke-Millson twist, and applies these results geometrically.

Findings

Injectivity of the Kudla-Millson lift under specified lattice conditions

Extension of known injectivity results to new lattice classes

Geometric applications to 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.