The mixing conjecture of Michel--Venkatesh can be thought of as an ergodic theoretic refinement of the André--Oort conjecture in the case of a product of modular curves. There are essentially two known approaches to this conjecture, both of which have yielded only conditional results. The first, due to Khayutin in a landmark paper, uses measure classification of higher rank diagonalizable actions and analytic number theory, but imposes splitting conditions on the discriminants and assumes no Siegel zeros. The other approach, due to Blomer, myself, and Khayutin, uses automorphic forms and analytic number theory, and most notably requires the generalized Riemann hypothesis (GRH). I will survey these results and methods, and describe ongoing work, joint with Blomer and Radziwiłł, in which we weaken the appeal to GRH in a closely related problem on simultaneous equidistribution to a more accessible hypothesis on the abundance of small split primes. The latter condition, when quantified, is of comparable strength to the absence of Siegel zeros, and allows us to capture more discriminants in the mixing conjecture than was known previously. |
 |
