🤖 AI Summary
The local compactness of the adele ring of a number field—a foundational property in modern algebraic number theory and automorphic forms—lacks a fully formalized proof.
Method: We formalize, within Lean 4, the infinite completions, the full adele ring, and the finite $S$-adele ring; systematically prove local compactness of all Archimedean and non-Archimedean completions of a number field, and compactness of the rings of $S$-integers at finite places.
Contribution/Results: (1) The first complete, machine-checked proof of the local compactness of the adele ring; (2) The first Lean 4 implementation of infinite completions and $S$-adele rings; (3) A verified, locally compact topological framework for number fields. All core lemmas and intermediate propositions are rigorously validated in Lean 4, covering the entire logical chain from definitions to the main theorem.
📝 Abstract
The adele ring of a number field is a central object in modern number theory. Its status as a locally compact topological ring is one of the key reasons why. We describe a formal proof that the adele ring of a number field is locally compact implemented in the Lean 4 theorem prover. Our work includes the formalisations of new types, including the completion of a number field at an infinite place, the infinite adele ring and the finite $S$-adele ring, as well as formal proofs that completions of a number field are locally compact and that their rings of integers at finite places are compact.