Weil-�tale cohomology over finite fields

Geisser, Thomas

doi:10.1007/s00208-004-0564-8

Cited by 36 publications

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References 14 publications

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“…A detailed version of the following discussion can be found in [6]. Let G ⊆ Gal(F q /F q ) be the Weil group, i.e.…”

Section: Arithmetic Cohomologymentioning

confidence: 99%

“…In [6], we calculated the precise relationship between Weil-etale cohomology groups and etale cohomology groups. In particular, if one assumes Tate's conjecture on the bijectivity of the cycle map and Beilinson's conjecture that rational and numerical equivalence agree up to torsion, then for smooth and projective varieties, the Weil-etale cohomology groups of the motivic complex have all properties expected by Lichtenbaum, and allow a new interpretation of results of Kahn [15].…”

Section: Introductionmentioning

confidence: 99%

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Arithmetic cohomology over finite fields and special values of ζ-functions

Geisser

2006

Duke Math. J.

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Summary. We construct cohomology groups with compact support H i c (Xar, Z(n)) for separated schemes of finite type over a finite field, which generalize Lichtenbaum's Weil-etale cohomology groups for smooth and projective schemes. In particular, if Tate's conjecture holds, and rational and numerical equivalence agree up to torsion, then the groups H i c (Xar, Z(n)) are finitely generated, form an integral model of ladic cohomology with compact support, and admit a formula for the special values of the ζ-function of X.

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“…A detailed version of the following discussion can be found in [6]. Let G ⊆ Gal(F q /F q ) be the Weil group, i.e.…”

Section: Arithmetic Cohomologymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Arithmetic cohomology over finite fields and special values of ζ-functions

Geisser

2006

Duke Math. J.

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“…Let X be a scheme of finite type over a finite field. In this subsection we use the Weil-étale topology W on X as defined by Lichtenbaum in [11] and studied by Geisser in [9]. In particular, we recall that there exists a natural morphism of topoi γ from the Weil-étale topos X W on X to theétale topos Xé t on X.…”

Section: 2mentioning

confidence: 99%

“…3.2] shows that RΓ(Xé t , F) belongs to D ct . Next, we recall from [9,Th. 3.3] that there exists a natural morphism θ : RΓ(Xé t , F) → RΓ(X W , γ * (F)) in D and that if D is any complex which lies in an exact triangle in D of the form In each degree i condition (i) combines with the exact cohomology sequence of (10) to give a short exact sequence 0…”

Section: 2mentioning

confidence: 99%

Perfecting the Nearly Perfect

Burns¹

2008

Pure and Applied Mathematics Quarterly

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We introduce a natural variant of the notion of nearly perfect complex. We show that this variant gives rise to canonical perfect complexes and prove several useful properties of this construction (including additivity of the associated Euler characteristics on suitable exact triangles). We then apply this approach to complexes arising from theétale cohomology of G m on arithmetic surfaces and discuss links to Lichtenbaum's theory of Weil-étale cohomology.

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“…As h 1 (E) has rank two, it is 2ρ X and c 2 X that arise rather than ρ X and c X . Note that the Weil-étale motivic cohomology groups H * W (X, Z X (n)) are known to be finitely generated only for n = 0 [12, §8]; the case n = 0 requires the Tate conjecture [5].…”

Section: Introductionmentioning

confidence: 99%

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Ramachandran¹

2005

Internat. Math. Res. Notices

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Abstract. We study the behaviour near s = 1 2 of zeta functions of varieties over finite fields Fq with q a square. The main result is an Euler-characteristic formula for the square of the special value at s = 1 2. The Euler-characteristic is constructed from the Weil-étale cohomology of a certain supersingular elliptic curve. IntroductionLet V be a integral scheme of finite type over Spec Z. The special values a V (n) at integers s = n of the zeta function ζ(V, s) of V are conjecturally related to deep arithmetical invariants of V . One may ask if the special values of ζ(V, s) at non-integral values of s, e.g. s = 1 2 , also admit an arithmetical interpretation. The simplest example -and perhaps the most interesting -arises from the central value of the Riemann zeta function ζ(s) = ζ(Spec Z, s):Is there a motivic interpretation of ζ( The motivic philosophy indicates that a V (n) depends on the interaction of the motive h(V ) of V with Z(±n) (power of the Tate motive). This leads us to suspect that the special value at s = 1 2 is governed by an exotic object (unknown to exist, as yet): a square root Z( It is clear that the investigation of special values at s = 1 2 should begin with the important case of varieties over finite fields. Namely, consider the zeta function ζ(X, s) of a smooth projective variety X over a finite field k = F q of characteristic p > 0. The function Z(X, t), defined by Z(X, q −s ) := ζ(X, s), is a rational function of t with integer coefficients. For any integer n ≥ 0, the order of the pole ρ n := −ord s=n ζ(X, s) at s = n and the special value a X (n) of Z(X, t) at t = q −n conjecturally admit a motivic interpretation.The Tate conjecture (Conjecture 2) predicts thatin the category of motives and h 2n (X) is part of the motive of X. A related variant is that ρ n is the rank of the Chow group CH n (X) of algebraic cycles of codimension n on X. The Lichtenbaum-Milne conjecture (Conjecture 3) expresses a X (n) as an Euler-characteristic ofétale motivic cohomology H * (X, Z X (n)); i.e., the cohomology of the (étale) motivic complexes Z X (n) of S. Lichtenbaum; Z X (0) is the constant sheaf Z and Z X (1) = G m [−1] is the sheaf G m in degree one. Their conjecture is known for n = 0 (unconditionally) and for n = 1 (modulo the Tate conjecture for divisors on X); cf. [11, 19]. For n = 0, 1, it takes the formLichtenbaum [12] has provided another elegant interpretation of a X (0) using his Weil-étale topology (cf. Theorem 5).Let us now turn to the special value at s = 1 2 or t = 1/ √ q. One can ask for a motivic description of • the order of vanishing ρ X := ord s= 1 2 ζ(X, s) and • the corresponding special value c X at s = 1 2 , viz., c X := lim t→1/ √ q (1 − √ qt) −ρX Z(X, t). The main result of this paper provides such a description, under the condition that q = p 2f , i.e., that F p 2 ⊂ F q . We note that c X may not be rational, if the condition on q is dropped.Our paper is an exploration, using the methods of [11,19,12,21], of a beautiful suggestion of Yuri Manin that "a certain supersingu...

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Weil-�tale cohomology over finite fields

Cited by 36 publications

References 14 publications

Arithmetic cohomology over finite fields and special values of ζ-functions

Arithmetic cohomology over finite fields and special values of ζ-functions

Perfecting the Nearly Perfect

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