2-cohomology of semi-simple simply connected group-schemes over curves defined over \(p\)-adic fields (Q394262)

Suppose \(k\) is finite extension of \(\mathbb{Q}_p\). Let \(X\) be a projective, smooth, geometrically irreducible curve over \(k\). Using a result of \textit{S. Lichtenbaum} [Invent. Math. 7, 120--136 (1969; Zbl 0186.26402)], namely, a perfect duality NEWLINE\[NEWLINE\mathrm{Br}(X) \times \mathrm{Pic}(X) \to \mathbb{Q}/\mathbb{Z},NEWLINE\]NEWLINE the author is able to prove that for any semi-simple simply connected \(X\)-group \(\tilde{G}\) and any \(X\)-band \(\mathcal{L}\) that is locally representable by \(\tilde{G}\), each of \(H^2_{\text{ét}}(X,\mathcal{L})\) is neutral.NEWLINENEWLINEUsing a higher dimensional generalization of [loc. cit.], the author is able to prove that if \(X\) is a projective, smooth, geometrically irreducible scheme of dimension greater than \(1\) over \(k\). Then each class of \(H^2_{\text{ét}}(X,\mathcal{L})/H^2_{\text{ét}}(\mathfrak{X},\mathcal{L})\) is natural, where \(\mathfrak{X}\) is a model of \(X\) over the ring \(\mathcal{O}_k\) of integers of \(k\), \(\mathcal{L}\) is a \(\mathfrak{X}\)-band which is locally representable by a scheme of semi-simple simply connected groups, and the order of \(Z(\mathcal{L})\) and \(p\) is coprime.