This is the last of four lectures given about "Quadratic Forms and Automorphic Forms" at the 2009 Arizona Winter School, to an audience of interested graduate students. Detailed notes supporting these lectures are available at: arxiv.org/abs/1105.5759
This (more advanced) lecture introduces the theta lifting, which is a way of passing between automorphic forms on symplectic and orthogonal groups. Here we construct the theta lifting from an explicit presentation of the (adelic) Weil representation. This is used to give a high-brow prove that the classical theta series (from Lecture 2) is a modular form.