Joint Linnik problems

We prove a conjecture of Michel--Venkatesh on joinings of distinct Linnik problems, in the setting of simultaneous quaternionic embeddings of imaginary quadratic fields having sufficiently many small split primes. This splitting condition is known to hold for all but $O((\log\log X)^{1 + o(1)})$ discriminants up to $X$. We also treat a non-equivariant form of this conjecture proposed by Aka--Einsiedler--Shapira, which in particular applies to the classical Gauss construction joining Linnik points on the sphere with CM points on the modular surface.