Congruences for the number of rational points over finite fields

The deep philosophical analogy between Hodge weights and weights for the Frobenius action on l-adic cohomology has  determined a large part of the development of algebraic geometry in the last 30 years. Less considered has been the analogy between Hodge and eigenvalue divisibility levels, a study we started in 2002. It particular it led to a positive solution of the Lang-Manin conjecture. We plan  an extension of the methods to non-smooth varieties,  and, if possible, to higher levels as well.