Decomposition theorem for abelian fibrations

Derived direct image of a proper map with smooth source is a direct sum of simple perverse sheaves with shifts in the degrees. The supports of these simple perverse sheaves are obviously important  topological invariants of the map. In general, it is difficult to determine these supports. This is possible for an abelian fibration under some assumptions. This determination has some amazing  consequences on equality of number of points of certain algebraic varieties over finite fields and in particular, it implies the so called fundamental lemma in Langlands' program.