Some consequences of the Riemann hypothesis for varieties over finite fields

Abstract

Abstract. We deduce from Deligne's form of the Riemann hypothesis and the hard Lefschetz theorem in r cohomology the corresponding facts for any "reasonable'" cohomology theory, in particular for crystalline cohomology, and give some applications to algebraic cycles.

Nilpotence of Frobenius action and the Hodge filtration on local cohomology

“…Therefore, the trace of the crystalline Frobenius on H 2 cris (X ν /W (k ν ))⊗Q p , which is a rational integer, is divisible by p. On the other hand, by Katz-Messing theorem [21], the characteristic polynomials of the crystalline Frobenius on H 2 cris (X ν /W (k ν ))⊗Q p and of the ℓ-adic Frobenius F ν on H 2 (Xé t , Q ℓ ) are equal to each other. Thus, the trace t ν of the ℓ-adic Frobenius F ν is divisible by p.…”

Geometry of Fermat adeles

“…Here i is a fixed integer, X y := X ⊗ κ(y), and H i is either the etale (l−adic, l prime to N (y)) or the crystalline cohomology tensored with Q. (The etale and crystalline theories give the same L−functions by [18].) 2) (Artin situation) We assume X/Y is a finite Galois cover of integral normal schemes, with Galois group G, and we assume we are given a finite-dimensional complex vector space V and a representation ρ : G → GL(V ) such that ρ is integral over K (i.e.…”

On the interior motive of certain Shimura varieties: the case of Picard surfaces

“…As for (c), in order to apply [KM,Thm. 2. 2)], use that both realizations are cut out by the same cycle from the cohomology of a smooth and proper scheme over the residue field F p of p (cmp.…”