Localized Bost-Connes systems over global function fields: commensurability forbids type III_1, and the factor recovers the temperature

Over a global function field all norms are powers of q, so the Radon-Nikodym cocycle takes values in a cyclic group: commensurability forbids type III_1, and the factor type recovers the inverse temperature. The compactification convention for the symmetry group in the degree direction is fixed, the detecting series of Picard characters are computed with explicit constants, and the positive-density detection theorem (Theorem 5.4) is proved via the detecting series of Pic-characters.