Weil's Rosetta Stone as a correspondence of clocks, not of spaces

Abstract

Weil's "trilingual text" — number fields, curves over finite fields, Riemann surfaces — is usually read as a dictionary between objects: primes ↔ points, Galois groups ↔ fundamental groups, and in Mazur's 3-dimensional refinement, primes ↔ knots, Legendre symbol ↔ linking number mod 2, quadratic extensions ↔ double branched covers. I want to record a different reading. In each column the objects sit alongside a generator of change: Frobenius on the arithmetic side, monodromy along loops (in Mazur's row, the meridian around a knot) on the topological side. Every successful transfer across the stone that I know of — the Weil conjectures via Frobenius ↔ Lefschetz fixed points, Mazur's dictionary sending Frobenius to the meridian loop — is a matching of these generators, with the correspondence of spaces appearing as the invariant residue. The proposal is that Weil's correspondence is better formalized as a morphism of dynamical systems (space + endomorphism/flow, or space + action of a "time" group) than as a correspondence of spaces, and that the "text" of the stone is a statement about orbits.