THE WEIL-´ETALE FUNDAMENTAL GROUP OF A NUMBER FIELD I

The Weil-étale fundamental group of a number field I 135
• There exists a fundamental class θ ∈ H 1 ( ¯X L , ˜R). The complex of finite-dimensional
vector spaces
···→Hc n−1 (X L , ˜R) → Hc n (X L, ˜R) → Hc
n+1 (X L , ˜R) →···
defined by cup product with θ, is acyclic.
• The vanishing order of the Dedekind zeta function ζ F (s) at s = 0 is given by
ord s=0 ζ F (s) = ∑ n≥0(−1) n n dim R H n c (X L, ˜R).
• The leading term coefficient ζF ∗ (s) at s = 0 is given by the Lichtenbaum Euler
characteristic
ζF ∗ (s) =±∏ |H n ( ¯X et ,τ ≤2 Rγ ∗ (ϕ ! Z)) tors | (−1)n /det(Hc n (X L, ˜R), θ, B ∗ ),
n≥0
where B n is a basis of H n ( ¯X et ,τ ≤2 Rγ ∗ (ϕ ! Z))/tors.
In particular, those results hold for the Weil-étale topos ¯X W defined in Section 6.
Proof. This follows from Theorems 6.5, 6.10, 6.11, Corollary 6.12 and from the analytic
class number formula.
✷
6.3. The sheaf R 2 γ ∗ Z
The étale sheaf R 2 γ ∗ Z is the sheaf associated to the presheaf
P 2 Z : Et ¯X −→ Ab
Ū ↦−→ (C 1 Ū )D .
Recall that if Ū is connected of function field K(Ū),thenCŪ is the S-idèle class group of
K(Ū),whereS is the set of places of K(Ū)not corresponding to a point of Ū.Inotherwords,
if we set K = K(U) then CŪ = C K,S is the S-idèle class group of K defined by the exact
sequence
∏
O
K × v
→ C K → C K,S → 0.
v∈U
The compact group C 1 Ū is then defined as the kernel of the canonical map C Ū → R× .Note
that such a finite set S does not necessarily contain all the archimedean places. The restriction
maps of the presheaf P 2 Z are induced by the canonical maps C ¯V → C Ū (well defined for any
étale map ¯V → Ū of connected étale ¯X-schemes). By class field theory, we have a covariantly
functorial exact sequence of compact topological groups
0 → D 1 Ū → C1 Ū → π 1(Ū et ) ab → 0,
where π 1 (Ū et ) ab is the abelian étale fundamental group of Ū and D 1 is the connected
Ū
component of 1 in C 1 Ū .Hereπ 1(Ū) ab is defined as the abelianization of the profinite
fundamental group of the Artin–Verdier étale topos ¯X et /yŪ ≃ Ū et . If we denote the function