The classical Linnik problems are concerned with the equidistribution of adelic torus orbits on the homogeneous spaces attached to inner forms of GL2, as the discriminant of the torus gets large. When specialized, these problems admit beautiful classical interpretations, such as the equidistribution of integer points on spheres, of Heegner points or packets of closed geodesics on the modular surface, or of supersingular reductions of CM elliptic curves. In the mid 20th century, Linnik and his school established the equidistribution of many of these classical variants through his ergodic method, under a congruence condition on the discriminants modulo a fixed auxiliary prime. More recently, the Waldspurger formula and subconvex estimates on L-functions were used to remove these congruence conditions, and provide effective power-savings rates.
In their 2006 ICM address, Michel and Venkatesh proposed a variant of this problem in which one considers the product of two distinct inner forms of GL2, along with a diagonally embedded torus. One can again specialize the setting to obtain interesting classical reformulations, such as the joint equidistribution of integer points on the sphere, together with the shape of the orthogonal lattice. This hybrid context has received a great deal of attention recently in the dynamics community, where, for instance, the latter problem was solved by Aka, Einsiedler, and Shapira, under supplementary congruence conditions modulo two fixed primes, using as critical input the joinings theorem of Einsiedler and Lindenstrauss.
In joint (ongoing) work with Valentin Blomer, we remove the supplementary congruence conditions in the joint equidistribution problem, conditionally on the Riemann hypothesis, while obtaining a logarithmic rate of convergence. The proof uses Waldsurger’s theorem and estimates of fractional moments of L-functions in the family of class group twists.
