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Let $M/L/Qp$ be a finite galois abelian extension of local fields and define

$\mathcal{M}=M\{\{T\}\}=\{\sum_{i\in \mathbb{Z}}a_iT^i:a_i\in M,\min_{i\in \mathbb{Z}}, v(a_i)>−\infty , \lim_{i\to −\infty }v(a_i)=\infty\}$.

Then $\mathcal{M}/\mathcal{L}$, $\mathcal{L}=L\{\{T\}\}$, is also a finite abelian galois extension. Then Kato's reciprocity map induces an isomorphism

$\Upsilon_L:K_2(\mathcal{L})/N(K_2(\mathcal{M}))→Gal(\mathcal{M}/\mathcal{L})$,

where $K_2(\mathcal{L})$ is the $2^{nd}$ Milnor $K$-group of $\mathcal{L}$ and $N:K_2(\mathcal{M})→K_2(\mathcal{L})$ is the norm on $K$-Milnor groups.

Since clearly

\begin{align*} Gal(M/L)&\cong Gal(\mathcal{M}/\mathcal{L})\\ g&\to g(\sum a_iT^i)=\sum g(a_i)T^i \end{align*}

then my question is, what it is the relation between the 2-dimensional reciprocity map ΥL and the 1-dimensional reciprocity map $\theta_L:L^∗\to Gal(M/L)$ in this scenario? I believe that a relation like

$\Upsilon_\mathcal{L}({a,T})=\theta_L(a)$, $a\in L^*$

should be true. I would like to know any reference or suggestion about this.

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    $\begingroup$ "Since clearly $\operatorname{Gal}(M/L)\cong\operatorname{Gal}(M/L)$..." Your identification $M = M\{\{T\}\}$ is confusing. I think you'd help your question by using $\tilde M$ or $M'$ to represent $M\{\{T\}\}$ instead of $M$, seeing as $M$ already has meaning. $\endgroup$
    – Stahl
    Oct 7, 2014 at 15:58
  • $\begingroup$ @Stahl Maybe you are on a phone but he is using script M to denote the power series field over M. $\endgroup$ Oct 7, 2014 at 22:24
  • $\begingroup$ I can see the script M now (I am now on my phone), so I think it was originally just M, but either way I'm seeing the script M now so I have no complaints! $\endgroup$
    – Stahl
    Oct 7, 2014 at 22:32

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