In the 90's it was observed that the shifted convolution sum $S(X,m)=\sum_{n\leq X}r(n)r(n+m)$ is closely related to a certain hyperbolic lattice point count. Based on this observation F. Chamizo (1999) established an asymptotic formula for $S(X,m)$ with high uniformity in $m$. However, his main result is conditional on an upper bound for a spectral moment of automorphic forms. In this talk we will revisit Chamizo's results and discuss how they can be made unconditional.

Homogeneous spaces arising as quotients of products of real and p-adic Lie groups by arithmetic lattices form a fruitful playground for the interaction between arithmetic and dynamics. Periodic orbits on such spaces are naturally related to classical arithmetic objects, including closed geodesics, Heegner points, integer points on spheres, and CM elliptic curves together with their reductions.

In this talk, I will explain how the ergodic-theoretic classification of joinings on homogeneous spaces, due to Einsiedler and Lindenstrauss, provides a tool for studying natural joint distribution problems involving these objects. I will also indicate how, in certain cases, this perspective suggests natural couplings between seemingly different arithmetic phenomena.