# From monoids to hyperstructures: in search of an absolute arithmetic

@article{Connes2010FromMT, title={From monoids to hyperstructures: in search of an absolute arithmetic}, author={Alain Connes and Caterina Consani}, journal={arXiv: Algebraic Geometry}, year={2010} }

We show that the trace formula interpretation of the explicit formulas expresses the counting functionN.q/ of the hypothetical curveC associated to the Riemann zeta function, as an intersection number involving the scaling action on the adele class space. Then, we discuss the algebraic structure of the adele class space both as a monoid and as a hyperring. We construct an extension R convex of the hyperfield S of signs, which is the hyperfield analogue of the semifield R max of tropical…

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