Tate's question, Standard conjecture D, semisimplicity and Dynamical degree comparison conjecture

2025/03/12 by Hu, Fei, Truong, Tuyen Trung, Xie, Junyi · 1 citation
#Algebraic Geometry (math.AG) #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2503.09432

Abstract

Let X be a smooth projective variety of dimension n over the algebraic closure of a finite field \mathbbF_p. Assuming the standard conjecture D, we prove a weaker form of the Dynamical Degree Comparison conjecture; equivalence of semisimplicity of Frobenius endomorphism and of any polarized endomorphism (a more general result, in terms of the biggest size of Jordan blocks, holds). We illustrate these results through examples, including varieties dominated by rational maps from Abelian varieties and suitable products of K3 surfaces. Using the same idea, we provide a new proof of the main result in a recent paper by the third author, including Tate's question/Serre's conjecture that for a polarized endomorphism f:X→ X, all eigenvalues of the action of f on H^k(X) have the same absolute value.

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• Standard conjecture D and some conjectures around Weil's Riemann hypothesis

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