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.