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.
