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<p>Let <i>f</i> be a dynamical correspondence of a smooth projective variety <i>X</i> over an algebraically closed field of arbitrary characteristic. In 2016, Truong conjectured that the dynamical degrees of <i>f</i> defined via the pullback actions on <i>ℓ</i>-adic étale cohomology groups and on numerical cycle class groups are equivalent, which we call the dynamical degree comparison (DDC) conjecture. It contains the generalized Weil’s Riemann hypothesis (for polarized endomorphisms) as a particular case. In this paper, we introduce a quantitative strengthening of the standard conjecture <i>C</i>, named as Conjecture <i>G</i><sub><i>r</i></sub>, and show that it holds on abelian varieties and Kummer surfaces. We prove that Conjecture <i>G</i><sub><i>r</i></sub> yields the generalized Weil’s Riemann hypothesis. Moreover, Conjecture <i>G</i><sub><i>r</i></sub> plus the standard conjecture <i>D</i> imply the norm comparison conjecture, whose consequences include the DDC conjecture and the generalized semisimplicity conjecture. As applications, we prove the DDC conjecture for algebraically stable dynamical correspondences of abelian varieties and Kummer surfaces, extending Hu’s previous work (2019, 2024) on endomorphisms of abelian varieties. We also obtain a similar comparison result for effective finite correspondences of abelian varieties and prove the generalized semisimplicity conjecture for Kummer surfaces.</p>

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A dynamical approach to generalized Weil’s Riemann hypothesis and semisimplicity

  • 期刊论文

  • 发表日期：

    2025年2月6日

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  • Fei Hu,
  • Tuyen Trung Truong

摘要

Let f be a dynamical correspondence of a smooth projective variety X over an algebraically closed field of arbitrary characteristic. In 2016, Truong conjectured that the dynamical degrees of f defined via the pullback actions on ℓ-adic étale cohomology groups and on numerical cycle class groups are equivalent, which we call the dynamical degree comparison (DDC) conjecture. It contains the generalized Weil’s Riemann hypothesis (for polarized endomorphisms) as a particular case. In this paper, we introduce a quantitative strengthening of the standard conjecture C, named as Conjecture Gr, and show that it holds on abelian varieties and Kummer surfaces. We prove that Conjecture Gr yields the generalized Weil’s Riemann hypothesis. Moreover, Conjecture Gr plus the standard conjecture D imply the norm comparison conjecture, whose consequences include the DDC conjecture and the generalized semisimplicity conjecture. As applications, we prove the DDC conjecture for algebraically stable dynamical correspondences of abelian varieties and Kummer surfaces, extending Hu’s previous work (2019, 2024) on endomorphisms of abelian varieties. We also obtain a similar comparison result for effective finite correspondences of abelian varieties and prove the generalized semisimplicity conjecture for Kummer surfaces.