Weil-etale cohomology of arithmetic schemes

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The proposer has constructed a Grothendieck topology for varieties over finite fields such that, modulo standard conjectures, special values of zeta-functions can be computed as Euler characteristics (in a suitable sense) of certain motivic sheaves or complexes of sheaves. He now intends to work on the construction of a similar Grothendieck topology for schemes over number rings. The Riemann zeta-function (discovered by Riemann in the nineteenth century) and its generalizations express many deep relations between analysis and number theory. These relations should be consequences of a deep underlying topological structure. This project is an investigation into the properties which that structure would have to possess.