Grothendieck's standard conjecture of Lefschetz type over finite fields

TL;DR

This paper discusses Grothendieck's standard conjecture of Lefschetz type over finite fields, highlighting its forms, known results, and potential implications for algebraic geometry.

Contribution

It explains the strong form of the conjecture and explores its significant consequences, emphasizing its accessibility over finite fields.

Findings

Weak form $C$ proven over finite fields

Strong form $B$ has notable consequences

Conjecture may be the most approachable among standard conjectures

Abstract

Grothendieck's standard conjecture of Lefschetz type has two main forms: the weak form $C$ and the strong form $B$. The weak form is known for varieties over finite fields as a consequence of the proof of the Weil conjectures. This suggests that the strong form of the conjecture in the same setting may be the most accessible of the standard conjectures. Here, as an advertisement for the conjecture, we explain some of its remarkable consequences.