Standard conjecture of Hodge type for powers of abelian varieties

2025/10/24 by Agugliaro, Thomas
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2510.21562

Abstract

We prove that the standard conjecture of Hodge type holds for powers of abelian threefolds. Along the way, we also prove the conjecture for powers of simple abelian variety of prime dimension over finite fields, and in other related cases based on the notion of Frobenius rank of Lenstra-Zarhin. The main tool is a result comparing two real fiber functors on tannakian categories. A second tool is a new an explicit description of simple Lefschetz motives over finite fields, in terms of ''enriched'' Frobenius eigenvalues.

Citations

• Examples for the standard conjecture of Hodge type
• The Hilbert symbol in the Hodge standard conjecture
• The Hodge standard conjecture for self-products of K3 surfaces
• The numerical Hodge standard conjecture for the square of a simple abelian variety of prime dimension
• Eigenvalues of Frobenius Endomorphisms of Abelian varieties of low\n Dimension
• Polarizations and Grothendieck's Standard Conjectures

Related

• Hodge structures on abelian varieties of type III
• A Survey of the Hodge Conjecture for Abelian Varieties
• Examples for the standard conjecture of Hodge type
• Hodge classes on abelian varieties
• The Tate and Standard Conjectures for Certain Abelian Varieties