Unfolding and injectivity of the Kudla-Millson lift of genus 1

TL;DR

This paper develops a new method to analyze the Kudla-Millson lift of genus 1, enabling explicit Fourier expansion computation and establishing its injectivity, which could extend to more general cases.

Contribution

It introduces an unfolding technique for theta integrals that simplifies Fourier expansion calculation and provides a new proof of injectivity, facilitating future generalizations.

Findings

Fourier expansion of Kudla-Millson lift computed explicitly.

Injectivity of the Kudla-Millson lift proven using new unfolding method.

Method paves the way for proving injectivity in higher genus and signature cases.

Abstract

We unfold the theta integrals defining the Kudla-Millson lift of genus 1 associated to even lattices of signature (b,2), where b>2. This enables us to compute the Fourier expansion of such defining integrals and prove the injectivity of the Kudla-Millson lift. Although the latter result has been already proved by Bruinier, our new procedure has the advantage of paving the ground for a strategy to prove the injectivity of the lift also for the cases of general signature and of genus greater than 1.