Cohomological divisibility and point count divisibility

Hélène Esnault, Nicholas M. Katz

Research output: Contribution to journalArticle

2 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 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, ℚ) with compact supports are divisible by q κ as algebraic integers.

Original languageEnglish (US)
Pages (from-to)93-100
Number of pages8
JournalCompositio Mathematica
Volume141
Issue number1
DOIs
StatePublished - Jan 1 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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