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, leading to its widespread use within the Langlands Program. We describe a formal proof that the adele ring of a number field is locally compact 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 and the finite $S$-adele ring, as well as formal proofs that completions of a number field are locally compact and their rings of integers at finite places are compact.

翻译：数域的阿代尔环是现代数论的核心对象。其作为局部紧拓扑环的性质是其能在朗兰兹纲领中被广泛应用的关键原因之一。本文描述了在Lean 4定理证明器中关于数域阿代尔环局部紧性的形式化证明。我们的工作包含新类型的构造，包括数域在无穷位处的完备化与有限$S$-阿代尔环，同时给出了数域完备化是局部紧的、以及其在有限位处的整数环是紧的形式化证明。