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I would like to know if the Kummer pairing (or the analogue of the Hilbert Symbol) for a one dimensional group defined over the ring of integers of a higher-dimensional local field is continuous (with respect to the Parshin topology of a higher-dimensional local field). To contextualize more my question,

Let $K$ be a $d$-dimensional local field of mixed characteristic, i.e, Char(K)=0 and Char(K_{d-1})=p, with maximal ring of integers $O_K$ and $F$ a one dimensional formal group over $O_K$. Let $L$ be a finite extension of $K$ containing the $n$-torsion points, $E_n$, of $F$ and let $\mu_L$ be the maximal ideal of $L$. Define the pairing

$K_d(L)\times \mu_L\to E_n :$ $(a, x)_{L,n}=\psi(a) (Z)-_FZ$

Where $K_d(L)$ is the Milnor $K$-group of $L$ and $[p^n](Z)=x$, here $[p^n]$ corresponds to multiplication-by-$p$ in the formal group $F$.

Let us fix an $a\in K_d(L)$ and consider the map $(a,)_{L,n}:\mu_L\to E_n$. Is this map continuous?

I appreciate any exact reference you may know about this issue.

Thanks!

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  • $\begingroup$ What is $\psi$? $\endgroup$ – David Loeffler Mar 17 '14 at 7:33
  • $\begingroup$ $\psi$ is Kato's reciprocity map. $\endgroup$ – user48406 Mar 17 '14 at 20:18

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