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For $K =\mathbb Q$, class field theory (or, in this case, Kronecker-Weber) says that the abelianization of the Galois group of $\mathbb Q$ is $\mathbb I/\mathbb I_\infty$. So a character $\operatornam …

since each idele has a divisor, the divisor is effective if and only if the idele is integral (locally at each place), the degree of that divisor equals the degree of the idele, and the measure of the adeles …

Observe that, because we can cover by open sets, on each of which the function is constant under translation by some open in the finite adeles, by compactness we can take a finite subcover, do the function …

I'm not sure if this is actually "spelled out", but here is a characterization of unstable vector bundles. First recall the degree map $GL_1(\mathbb A) \to \mathbb Z$ which sends an idele to the sum …

Then we will replace the adeles with a product over the points actually in $X$, and do the same for the integral ideles. Then let's make this set into a category. …