On the Weil-\'etale topos of regular arithmetic schemes

TL;DR

This paper constructs a Weil-étale topos for regular, proper schemes over Spec(Z), linking its cohomology to zeta functions and L-functions, extending previous work by Lichtenbaum and Geisser.

Contribution

It defines a Weil-étale topos for regular schemes over Spec(Z) with properties aligning with conjectured relations to zeta functions, generalizing prior cohomological frameworks.

Findings

Cohomology with R-coefficients relates to ζ(X,s) at s=0 under certain L-function conditions.

Cohomology with Z-coefficients matches previous theories in characteristic p cases.

The topos satisfies properties suggested by Lichtenbaum for such structures.

Abstract

We define and study a Weil-\'etale topos for any regular, proper scheme $X$ over $\Spec(Z)$ which has some of the properties suggested by Lichtenbaum for such a topos. In particular, the cohomology with $R$-coefficients has the expected relation to $\zeta(X,s)$ at $s=0$ if the Hasse-Weil L-functions $L(h^i(X_Q),s)$ have the expected meromorphic continuation and functional equation. If $\X$ has characteristic $p$ the cohomology with $Z$-coefficients also has the expected relation to $\zeta(X,s)$ and our cohomology groups recover those previously studied by Lichtenbaum and Geisser.