Cohomological divisibility and point count divisibility

Hélène Esnault; Nicholas M. Katz

doi:10.1112/S0010437X04000934

Cohomological divisibility and point count divisibility

Hélène Esnault, Nicholas M. Katz

Mathematics

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3 Scopus citations

Abstract

Let X ⊂ ℙⁿ be a closed scheme defined by r homogeneous equations of degrees d₁ ≥ d₂ ≥ ⋯ ≥ d _r over the finite field double-struck F sign_q, with complement U := ℙⁿ\X. Let κ be the maximum of 0 and the integral part of the rational number (n - d₂ - ⋯ - d _r)/d₁. We show that the eigenvalues of the geometric Frobenius endomorphism acting on the ℓ-adic cohomology Hⁱ _c(U ×_Fq double-struck F sign̄_q, ℚ_ℓ) with compact supports are divisible by q ^κ as algebraic integers.

Original language	English (US)
Pages (from-to)	93-100
Number of pages	8
Journal	Compositio Mathematica
Volume	141
Issue number	1
DOIs	https://doi.org/10.1112/S0010437X04000934
State	Published - Jan 2005

All Science Journal Classification (ASJC) codes

Algebra and Number Theory

Keywords

Ax-Katz theorem
Eigenvalues of Frobenius
Rational points over finite fields

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N2 - Let X ⊂ ℙn be a closed scheme defined by r homogeneous equations of degrees d1 ≥ d2 ≥ ⋯ ≥ d r over the finite field double-struck F signq, with complement U := ℙn\X. Let κ be the maximum of 0 and the integral part of the rational number (n - d2 - ⋯ - d r)/d1. We show that the eigenvalues of the geometric Frobenius endomorphism acting on the ℓ-adic cohomology Hi c(U ×Fq double-struck F sign̄q, ℚℓ) with compact supports are divisible by q κ as algebraic integers.

AB - Let X ⊂ ℙn be a closed scheme defined by r homogeneous equations of degrees d1 ≥ d2 ≥ ⋯ ≥ d r over the finite field double-struck F signq, with complement U := ℙn\X. Let κ be the maximum of 0 and the integral part of the rational number (n - d2 - ⋯ - d r)/d1. We show that the eigenvalues of the geometric Frobenius endomorphism acting on the ℓ-adic cohomology Hi c(U ×Fq double-struck F sign̄q, ℚℓ) with compact supports are divisible by q κ as algebraic integers.

KW - Ax-Katz theorem

KW - Eigenvalues of Frobenius

KW - Rational points over finite fields

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Cohomological divisibility and point count divisibility

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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