Theorem 2.2.3 (000A): Discreteness and cocompactness

Slogan: The rational numbers sit in the adeles the way the integers sit in the real numbers.

Reference: [Bump, Section 3.1]

Theorem 2.2.3 (Discreteness and cocompactness). The subgroup \(\mathbb{Q}\subset \mathbb{A}\) is discrete, and the quotient \(\mathbb{A}/ \mathbb{Q}\) is compact.

Proof. By (2.2.2.1) the open neighbourhood \((-1, 1) \times \prod_p \mathbb{Z}_p\) of \(0\) meets \(\mathbb{Q}\) only in \(0\), and \(\mathbb{A}/ \mathbb{Q}\) is the image of the compact set \([0, 1] \times \prod_p \mathbb{Z}_p\).□