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{\Large Department of Mathematics, BGU}

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\textbf{On} \emph{Wednesday, November 13, 2024}
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\textbf{At} \emph{14:10 -- 15:10}
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\textbf{In} \emph{-101}

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{\large\scshape Mikhail Borovoi 
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  (TAU)
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will talk about
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{\Large\bfseries The power operation in the Galois cohomology of a reductive group over a number field\par}
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\textsc{Abstract:}
For a number field K admitting a real embedding, it is impossible to construct a functorial in G group structure in the Galois cohomology pointed set H\^{}1(K,G) for all connected reductive K-groups G. However, over an arbitrary number field K, we define a \emph{diamond} (or \emph{power}) operation of raising to power n
         (x,n) \textbackslash{}mapsto x\^{}\{\textbackslash{}Diamond n\}:  H\^{}1(K,G) \textbackslash{}times Z -\textgreater{} H\^{}1(K,G).
We show that this operation has many functorial properties. When G is a torus, the set H\^{}1(K,G) has a natural group structure, and x\^{}\{\textbackslash{}Diamond n\} coincides with the n-th power of  x in this group.

For a cohomology class x in H\^{}1(K,G), we  define the period per(x) to be the greatest common divisor of n\textgreater{}0 such that x\^{}\{\textbackslash{}Diamond n\}=1, and the index ind(x) to be the greatest common divisor of the degrees {[}L:K{]} of finite separable extensions L/K splitting x.  These period and index generalize the period and index of a central simple algebra over K (in the special case where G is the projective linear group PGL\_n, the elements of H\^{}1(K, G) can be represented by central simple algebras). For an arbitrary reductive group G defined over a local or global field K, we show  that  per(x) divides ind(x), that per(x) and ind(x) have the same prime factors, but the equality  per(x)=ind(x) may not hold.

The talk is based on a joint work with Zinovy Reichstein.








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