Local-Global Principles and Effective Rates of Equidistribution For Semisimple Orbits

We prove an effective equidistribution theorem for semisimple

closed orbits on compact adelic quotients. The obtained error depends polynomially on the minimal complexity of intermediate orbits and the complexity of the ambient space. As an application, we establish a local-global principle for representations of quadratic forms, improving the codimension assumptions and providing effective bounds in a theorem of Ellenberg and Venkatesh. This is joint work with Manfred Einsiedler, Elon Lindenstrauss, and Amir Mohammadi.