Formalizing the Ring of Adèles of a Global Field

The ring of adèles of a global field and its group of units, the group of idèles, are fundamental objects in modern number theory. We discuss a formalization of their definitions in the Lean 3 theorem prover. As a prerequisite, we formalize adic valuations on Dedekind domains. We present some applications, including the statement of the main theorem of global class field theory and a proof that the ideal class group of a number field is isomorphic to an explicit quotient of its idèle class group. Acknowledgements I would like to thank Kevin Buzzard for his constant support and for many helpful conversations during the completion of this project, and Ashvni Narayanan for pointing out that the finite adèle ring can be defined for any Dedekind domain. I am also grateful to Patrick Massot for making some of the topological prerequisites available in mathlib , and to Sebastian Monnet for formalizing the topology on the infinite Galois group. Finally, I thank the mathlib community for their helpful advice, and the mathlib maintainers for the insightful reviews of the parts of this project already submitted to the library.