Towards a Geometric Analogue of Sarnak's Conjecture

Work of Mark Shusterman and myself has proven an analogue of Chowla's conjecture for polynomial rings over finite fields, which controls k-points correlations of the Möbius function for k bounded by a certain function of the finite field size. Sarnak has shown that Chowla's conjecture implies Sarnak's conjecture on correlations of the Möbius function with zero-entropy dynamical systems. However, our analogue is not quite strong enough to imply an analogue of Sarnak's conjecture, since that requires control on k-point correlations for all k. I will propose a geometric version of Sarnak's conjecture to which our methods do apply.