Twisted correlations of the divisor function via discrete averages of SL₂(R) Poincar\'e series

We prove a theorem that allows one to count solutions to determinant equations twisted by a periodic weight with high uniformity in the modulus. It is obtained by using spectral methods of SL₂(R) automorphic forms to study Poincar\'e series over congruence subgroups while keeping track of interactions between multiple orbits. In this way we get advantages over the widely used sums of Kloosterman sums techniques. We showcase the method with applications to correlations of the divisor functions twisted by periodic functions and the fourth moment of Dirichlet L-functions on the critical line.