Joint IAS/PU Number Theory Seminar

The linearization method of Dani-Margulis controls the amount of time a unipotent trajectory spends near invariant subvarieties of a homogeneous space. I will describe a problem in number theory where a similar control is desired for diagonalizable torus actions, but no analog of linearization holds. I will present two alternative approaches. The first one draws on measure rigidity for higher rank diagonalizable actions. The second approach is new joint work with V. Blomer and F. Brumley. This method uses the relation between torus periods on GL2 and L-functions. The two are ostensibly very different, but they share a common underlying structure, and both rely crucially on sieve methods.