On the Weil-étale cohomology of number fields

Morin, Baptiste

doi:10.1090/s0002-9947-2011-05124-x

Cited by 6 publications

(18 citation statements)

References 7 publications

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“…where RZ is the complex defined in [38] Theorem 8.5 and R W Z := RZ ≤2 . The complex RΓ(X W , Z), defined in [35], is the cohomology of the Weil-étale topos X W which is defined in [39].…”

Section: 5mentioning

confidence: 99%

Zeta functions of regular arithmetic schemes at s=0

Morin¹

2014

Duke Math. J.

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In [35] Lichtenbaum conjectured the existence of a Weil-étale cohomology in order to describe the vanishing order and the special value of the Zeta function of an arithmetic scheme X at s = 0 in terms of Euler-Poincaré characteristics. Assuming the (conjectured) finite generation of some étale motivic cohomology groups we construct such a cohomology theory for regular schemes proper over Spec(Z). In particular, we obtain (unconditionally) the right Weil-étale cohomology for geometrically cellular schemes over number rings. We state a conjecture expressing the vanishing order and the special value up to sign of the Zeta function ζ(X , s) at s = 0 in terms of a perfect complex of abelian groups RΓW,c(X , Z). Then we relate this conjecture to Soulé's conjecture and to the Tamagawa number conjecture of Bloch-Kato, and deduce its validity in simple cases. (2010): 14F20 · 14G10 · 11S40 · 11G40 · 19F27 Mathematics Subject Classification

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Section: 5mentioning

confidence: 99%

Zeta functions of regular arithmetic schemes at s=0

Morin¹

2014

Duke Math. J.

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“…There exist complexes R W (ϕ ! Z) and R W (Z) of sheaves on the Artin-Verdier étale topos whose hypercohomology is the conjectural Lichtenbaum cohomology with and without compact support respectively (see [15]). This suggests the existence of a canonical morphism of topoi…”

Section: 3mentioning

confidence: 99%

“…for any v not in U . For an ultrametric place v, the morphism [15] Prop. 6.2) and by a geometric point of X over v. If v is archimedean, G k(v) = {1} and u v : Sets → Xet is the point of the étale topos corresponding to v ∈ X.…”

Section: The Fundamental Group and Unramified Class Field Theorymentioning

confidence: 99%

The Weil-Étale Fundamental Group of a Number Field I

Morin¹

2011

Kyushu J. Math.

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Abstract. Lichtenbaum has conjectured (Ann of Math. (2) 170(2) (2009), 657-683) the existence of a Grothendieck topology for an arithmetic scheme X such that the Euler characteristic of the cohomology groups of the constant sheaf Z with compact support at infinity gives, up to sign, the leading term of the zeta function ζ X (s) at s = 0. In this paper we consider the category of sheavesX L on this conjectural site for X = Spec(O F ) the spectrum of a number ring. We show thatX L has, under natural topological assumptions, a welldefined fundamental group whose abelianization is isomorphic, as a topological group, to the Arakelov-Picard group of F . This leads us to give a list of topological properties that should be satisfied byX L . These properties can be seen as a global version of the axioms for the Weil group. Finally, we show that any topos satisfying these properties gives rise to complexes ofétale sheaves computing the expected Lichtenbaum cohomology.

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“…As observed in [11], the Weil-étale cohomology introduced in [8] is not defined as the cohomology of a Grothendieck site (i.e. of a topos).…”

Section: Introductionmentioning

confidence: 99%

“…This group is the Galois group of the maximal sub-extension ofF /K unramified at any place of K corresponding to ofŪ (regardless if such a place is ultrametric or archimedean). More geometrically, we consider the filtered set of pointed Galois étale cover {(V , qV ) → (Ū , qŪ )} to define the étale fundamental groupπ 1 (Ū et , qŪ ) := lim ←− (V ,qV ) Gal(V /Ū )The pair(11) (π 1 (Ū et , qŪ ), lim −→ (V ,qV ) CV ) is a (topological) class formation (see[14] Proposition 8.3.8 and [14] Theorem 8.3.12). This follows from the fact that if L/K is a Galois extension unramified overŪ , then the G L/K -module w∈V O × Lw in…”

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confidence: 99%

The Weil-étale fundamental group of a number field II

Morin

2010

Sel. Math. New Ser.

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Abstract. We define the fundamental group underlying to Lichtenbaum's Weil-étale cohomology for number rings. To this aim, we define the Weilétale topos as a refinement of the Weil-étale sites introduced in [8]. We show that the (small) Weil-étale topos of a smooth projective curve defined in this paper is equivalent to the natural definition given in [7]. Then we compute the Weil-étale fundamental group of an open subscheme of the spectrum of a number ring. Our fundamental group is a projective system of locally compact topological groups, which represents first degree cohomology with coefficients in locally compact abelian groups. We apply this result to compute the Weil-étale cohomology in low degrees and to prove that the Weil-étale topos of a number ring satisfies the expected properties of the conjectural Lichtenbaum topos.

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On the Weil-étale cohomology of number fields

Cited by 6 publications

References 7 publications

Zeta functions of regular arithmetic schemes at s=0

Zeta functions of regular arithmetic schemes at s=0

The Weil-Étale Fundamental Group of a Number Field I

The Weil-étale fundamental group of a number field II

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