Standard conjectures for abelian fourfolds

TL;DR

This paper proves the Standard Conjecture of Hodge type for abelian fourfolds and uses p-adic Hodge Theory to relate numerical and l-adic homological equivalences, advancing understanding in algebraic geometry.

Contribution

It establishes the Standard Conjecture for abelian fourfolds and links numerical and l-adic homological equivalences via p-adic methods, a novel approach in the field.

Findings

Proved the Standard Conjecture of Hodge type for abelian fourfolds.

Demonstrated that numerical and l-adic homological equivalences coincide for infinitely many l.

Applied p-adic Hodge Theory to solve a longstanding problem in algebraic geometry.

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.