GAAG seminar - Joint Linnik Problems

A well-known class of arithmetic equidistribution problems, attributed to Linnik, is concerned with periodic toric orbits on quaternionic varieties. Classical examples include the equidistribution of CM points of large discriminant on the modular surface and projections to the sphere of integer solutions to the sum of three squares. These problems were essentially solved by Duke using techniques in automorphic forms and analytic number theory. One can combine any two Linnik problems using a diagonal action of the torus, which encodes their simultaneous equidistribution (or disjointness). This creates a new set of problems, first put forward by Michel and Venkatesh, of considerably greater difficulty. We present new work with Valentin Blomer and Maksym Radziwiłł which uses an array of automorphic and analytic number theoretic techniques to prove the simultaneous equidistribution of two distinct Linnik problems, under a no-Siegel-zero type hypothesis. The latter assumption encodes the abundance of small split primes in quadratic field extensions, a property which interacts directly with competing approaches emanating from ergodic theory.