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Eigenvalues and dynamical degrees of self-maps on abelian varieties

Fei Hu

Abstract

Let $X$ be a smooth projective variety over an algebraically closed field, and $f\colon X\to X$ a surjective self-morphism of $X$. The $i$-th cohomological dynamical degree $χ_i(f)$ is defined as the spectral radius of the pullback $f^{*}$ on the étale cohomology group $H^i_{\textrm{ét}}(X, \mathbf{Q}_\ell)$ and the $k$-th numerical dynamical degree $λ_k(f)$ as the spectral radius of the pullback $f^{*}$ on the vector space $\mathsf{N}^k(X)_{\mathbf{R}}$ of real algebraic cycles of codimension $k$ on $X$ modulo numerical equivalence. Truong conjectured that $χ_{2k}(f) = λ_k(f)$ for all $0 \le k \le \dim X$ as a generalization of Weil's Riemann hypothesis. We prove this conjecture in the case of abelian varieties. In the course of the proof we also obtain a new parity result on the eigenvalues of self-maps of abelian varieties in prime characteristic, which is of independent interest.

Eigenvalues and dynamical degrees of self-maps on abelian varieties

Abstract

Let be a smooth projective variety over an algebraically closed field, and a surjective self-morphism of . The -th cohomological dynamical degree is defined as the spectral radius of the pullback on the étale cohomology group and the -th numerical dynamical degree as the spectral radius of the pullback on the vector space of real algebraic cycles of codimension on modulo numerical equivalence. Truong conjectured that for all as a generalization of Weil's Riemann hypothesis. We prove this conjecture in the case of abelian varieties. In the course of the proof we also obtain a new parity result on the eigenvalues of self-maps of abelian varieties in prime characteristic, which is of independent interest.

Paper Structure

This paper contains 7 sections, 15 theorems, 76 equations, 1 table.

Table of Contents

  1. Introduction
  2. Parity of eigenvalues
  3. An application to algebraic dynamics
  4. Preliminaries on abelian varieties
  5. Proof of Theorem \ref{['thm:B']}
  6. Proof of Corollaries \ref{['cor:A']} and \ref{['cor:B']}
  7. Proof of Theorem \ref{['thm:A']}

Key Result

Theorem 1.1

Let $\alpha \colon X \to X$ be an endomorphism of an abelian variety $X$ of dimension $g$ over $\mathbf{k}$. Let $P_\alpha(t)$ be the characteristic polynomial of the pullback $\alpha^*$ on the first étale cohomology group $H^1_{\emph{\'et}}(X, \mathbf{Q}_\ell)$ of $X$, which is a monic polynomial o

Theorems & Definitions (39)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Corollary 1.4
  • Definition 1.5
  • Remark 1.6
  • Corollary 1.7
  • Conjecture 1.8: cf. Truong
  • Theorem 1.9
  • Remark 1.10
  • ...and 29 more