Standard conjectures for abelian fourfolds

Abstract

Let A be an abelian fourfold. We prove the Standard Conjecture of Hodge type for A. By combining this result with a theorem of Clozel we deduce that numerical equivalence on A coincides with l-adic homological equivalence on A for infinitely many l. The approach consists in reformulating this question into a p-adic problem and then using p-adic Hodge Theory to solve it.

Final version, published online in Inventiones Mathematicae. The p-adic part of the paper has been improved following ideas of the referee