Parity and symmetry of polarized endomorphisms on cohomology

TL;DR

The paper proves that for any $q$-polarized endomorphism $f$ on a smooth projective variety $X$, the eigenvalues of $f^*$ on $\ell$-adic étale cohomology satisfy parity and symmetry properties predicted by the standard conjectures, extending the Frobenius case via Xie’s Weil RH-type result. It establishes a functional equation $t^{b_i} P_i(q^i/t) = (-1)^{\epsilon_i} q^{ib_i/2} P_i(t)$ for the characteristic polynomials $P_i(t)$ and shows that the real eigenvalues $\pm q^{i/2}$ have even multiplicities when $0<i<2d$, with related consequences for Newton polygons. The argument combines a preserved bilinear form $B$ and a normalized operator $\varphi_i$, Katz–Messing integrality, and Xie’s results to deduce eigenvalue structure, and yields non-arch Newton-polygon bounds, including sharp cases for abelian varieties and Grassmannians. An explicit abelian-surface example illustrates the sharpness of the results and the Newton-polygon phenomena.

Abstract

We show that the eigenvalues of any polarized endomorphism acting on the $\ell$-adic étale cohomology of a smooth projective variety satisfy certain parity and symmetry properties, as predicted by the standard conjectures. These properties were previously known for Frobenius endomorphisms. Besides the hard Lefschetz theorem, a key new ingredient is a recent Weil's Riemann hypothesis-type result due to J.~Xie. We also prove a "Newton over Hodge" type property for abelian varieties and Grassmannians.

Theorems & Definitions (20)

• Conjecture 1.1: cf. Tate65
• Remark 1.2
• Theorem 1.3: cf. Xie-GWRH
• Theorem 1.4
• Remark 1.5
• Corollary 1.6
• Remark 1.8
• Proposition 1.9
• Remark 1.10
• Lemma 2.1