Knots, primes and class field theory

TL;DR

This paper extends classical class field theory into a geometric framework using adelic and scheme-theoretic constructions, providing new insights into the spectral realization of L-function zeros and Galois actions.

Contribution

It introduces a geometric generalization of class field theory that connects adelic constructions with schemes and the ring of integers, offering a conceptual understanding of the adele class space.

Findings

Constructs a functor from finite abelian extensions of Q to finite covers of the adele class space.

Shows monodromy of periodic orbits corresponds to Galois actions of Frobenius.

Provides a geometric interpretation of the role of the idele class group in spectral number theory.

Abstract

In this paper, we present a geometric generalization of class field theory, demonstrating how adelic constructions, central to the spectral realization of zeros of L-functions and the geometric framework for explicit formulas in number theory, naturally extend the classical theory. This generalization transitions from the idele class group, which acts as the adelic analog of Galois groups, to a geometric framework associated with schemes and the ring of integers of global fields. This perspective provides a conceptual explanation for the role of the adele class space in the spectral realization of L-function zeros and identifies the idele class group as a generic point in this context. The sector $X_{\mathbb{Q}}$ of the adele class space corresponding to the Riemann zeta function gives the class field counterpart of the scaling topos. The main result is the construction of a functor mapping finite abelian extensions of $\mathbb{Q}$ to finite covers of $X_{\mathbb{Q}}$, with the monodromy of periodic orbits of length $\log p$ under the scaling action corresponding to the Galois action of the Frobenius at the prime p.