The Bloch-Kato conjecture for good reduction curves over local fields (Q1806337)

The Bloch-Kato-Milnor conjecture asserts that, for an arbitrary field \(F\), the canonical homomorphism \(h^q_F:K^M_q (F)/n\to H^q(F, \mu_n^{\otimes q})\) induced by the cup-product pairing \((n\geq 0\), prime to \(\text{char}(F)\); \(q\geq 0)\) is actually an isomorphism. In this paper, the author proves this conjecture in the case \(n\) is a prime \(\ell\) and \(F\) is the function field of a smooth projective geometrically connected curve with good reduction over a local field. The case \(q=2\) is already settled by the Merkur'ev-Suslin theorem. For \(q\geq 2\), the Milnor \(K\)-group is divisible by \(\ell\) and the cohomology group is trivial, because of the cohomological dimension of \(K\). The remaining case \(q=3\) is proved by mimicking Tate's approach for number fields, taking advantage of arithmetic properties of the field such as Hasse principles and class field theory.