Abstract: Let X be a complex modular curve or a Shimura curve. The Galois orbit of a Heegner point x in X is related by the theory of complex multiplication to an orbit of a Picard group of a quadratic order. A special point on the Cartesian product X x X is a tuple of Heegner points (x,y).  If x and y are associated to the same quadratic order, then the Galois orbit of (x,y) in X x X is also an orbit of the Picard group, acting diagonally. Michel and Venkatesh have conjectured that a sequence of Galois orbits of such pairs becomes equidistributed in X x X with respect to the product of the hyperbolic measures, unless an obvious obstruction occurs. I have previously established this conjecture for sequences of Galois orbits with discriminants satisfying a congruence condition modulo two distinct primes, under the assumption of no Siegel zeros. There is also an analogous conjecture arising from definite inner forms of PGL2 about the equidistribution of diagonal orbits of Picard groups on lattice points on products of ellipsoids.

I will discuss joint work in progress with Farrell Brumley and Valentin Blomer where we establish an effective version of the mixing conjecture of Michel and Venkatesh under GRH and GRC for anisotropic forms of PGL2, without assuming the congruence assumption. Our general approach is via the Weyl equidistribution criterion, which we verify by combining three methods.  The first method, inspired by previous work by Ellenberg-Michel-Venkatesh, is valid when the shift between the points in the pair (x,y) is appropriately small. The second method, originally due to Blomer-Brumley, is valid for a product of two cusp forms lying in different automorphic representations. The final ingredient uses a sieve, and it works for a product of two cusp forms from the same representation. It is inspired by my previous result on the mixing conjecture. I will try and explain some ideas behind these methods, focusing on the third one.