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

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 in [5], 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.