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.
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory
- Ax-Katz theorem
- Eigenvalues of Frobenius
- Rational points over finite fields