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Let K be the fraction field of a two-dimensional henselian, excellent, and equicharacteristic local domain. We prove a local-global principle for Galois cohomology with certain finite coefficients over K. We use classical machinery from étale cohomology theory, drawing upon an idea in Saito's work on two-dimensional local class field theory. This approach works equally well over the function field of a curve over an equi-characteristic henselian discrete valuation field, thereby giving a different proof of (a slightly generalized version of) a recent result of Harbater, Hartmann, and Krashen. We also present two applications. One is the Hasse principle for torsors under quasisplit semisimple simply connected groups without E8 factor. The other gives an explicit upper bound for the Pythagoras number of a Laurent series field in three variables. This bound is sharper than earlier estimates. [ABSTRACT FROM AUTHOR]

We prove a simple explicit formula for the local Katz–Radon transform of an ℓ-adic representation of the Galois group of the fraction field of a strictly henselian discrete valuation ring with positive residual characteristic, which can be defined as the local additive convolution with a fixed tame character. The formula is similar to the one proved by Arinkin in the -module setting, and answers a question posed by Katz. [ABSTRACT FROM AUTHOR]

If the absolute Galois group GK of a field K is a direct product GK=G1 × G2, then one of the factors is prosolvable and either G1 and G2 have coprime order or K is Henselian and the direct product decomposition reflects the ramification structure of GK. So, typically, the direct product of two absolute Galois groups is not an absolute Galois group. In contrast, free (profinite) products of absolute Galois groups are known to be absolute Galois groups. The same is true about free pro-p products of absolute Galois groups which are pro-p groups. We show that, conversely, if C is a class of finite groups closed under forming subgroups, quotients, and extensions, and if the class of pro-C absolute Galois groups is closed under free pro-C products, then C is either the class of all finite groups or the class of all finite p-groups. As a tool, we prove a generalization of an old theorem of Neukirch which is of interest in its own right: if K is a non-Henselian field, then every finite group is a subquotient of GK, unless all decomposition subgroups of GK are pro-p groups for a fixed prime p. [ABSTRACT FROM AUTHOR]