## The Weil-étale topology for number rings.(English)Zbl 1278.14029

Summary: There should be a Grothendieck topology for an arithmetic scheme $$X$$ such that the Euler characteristic of the cohomology groups of the constant sheaf $$\mathbb Z$$ with compact support at infinity gives, up to sign, the leading term of the zeta-function of $$X$$ at $$s = 0$$. We construct a topology (the Weil-étale topology) for the ring of integers in a number field whose cohomology groups $$H^i(\mathbb Z)$$ determine such an Euler characteristic if we restrict to $$i \leq 3$$.

### MSC:

 14F20 Étale and other Grothendieck topologies and (co)homologies 11R34 Galois cohomology 14G50 Applications to coding theory and cryptography of arithmetic geometry 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)

### Keywords:

étale cohomology; zeta-function; Weil group
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### References:

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