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

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