PRIMES, KNOTS AND THE SCALING SITE

The scaling site and its periodic orbits of length log p provide a geometric construction where one can interpret the well-known analogy between primes and knots. The role of the maximal abelian cover of the scaling site is played by the adele class space of the rationals. The inverse image of a p-periodic orbit is canonically isomorphic to the mapping torus of the multiplication by the p-Frobenius in the abelianized étale fundamental group of the spectrum of the integers localized at p, thus exhibiting the link of p with the other primes. (Based on joint work with Alain Connes.)