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Let K be a complete discretely valued field of characteristic 0 with residue field κ of characteristic p. Let n = [κ : κ p ] be the p-rank of κ. It was proved in [PS14] that the Brauer p-dimension of K lies between n/2 and 2n. For n ≤ 3, we improve the upper bound to n + 1 and provide examples to show that our bound is sharp. For n ≤ 2, we also improve the lower bound to n. For general n, we construct a family of fields Kn with residue fields of p-rank n, such that Kn admits a central simple algebra Dn of index p n+1 . Our sharp lower bounds for n ≤ 2 and upper bounds for n ≤ 3 in combination with the nature of these examples motivate us to conjecture that the Brauer p-dimension of such fields always lies between n and n + 1.

Let k be a field of characteristic not 2. We give a positive answer to Serre's injectivity question for any smooth connected reductive k-group whose Dynkin diagram contains connected components only of type A n , B n or C n . We do this by relating Serre's question to the norm principles proved by Barquero and Merkurjev. We give a scalar obstruction defined up to spinor norms whose vanishing will imply the norm principle for the non-trialitarian D n case and yield a positive answer to Serre's question for connected reductive k-groups whose Dynkin diagrams contain components of non-trialitarian type D n also. We also investigate Serre's question for reductive k-groups whose derived subgroups admit quasi-split simply connected covers. arXiv:1511.00311v2 [math.AG]

Let F be the function field of a curve over a p-adic field. Let D/F be a central division algebra of prime exponent ℓ which is different from p. Assume that F contains a primitive ℓ 2 th root of unity. Then the abstract group SK 1 (D) :

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