For all that follows, $p$ is a fixed odd prime. In the formulation of the Noncommutative Main Conjecture of Iwasawa theory one uses étale cohomology to define an algebraic object analogous to Iwasawas 'charakteristic ideal' in $\Lambda(G)$ for $G=Gal(k_\infty/k)$, with $k^{cyc}\subset k_\infty$ and $\mu=0$:

Let $M_\Sigma$ be the maximal abelian, pro-$p$, outside of $\Sigma$ unramified extension of $k_\infty$ for a finite set of primes of $k$, $\Sigma$. Then $X:=Gal(M_\Sigma/k_\infty)$ is a $\Lambda(G)$ module. If $G$ has elements of order $p$ we are prevented from seeing this $X$ a relative $K_0$ group, associated to a denominator set, $S$, of $\Lambda(G)$.

Now étale cohomology enters the picture: We use it to define a complex $C$ of $\Lambda(G)$-modules which is $S$-acyclic and quasi-isomorphic to a bounded complex of finitely generated $\Lambda(G)$-modules. Although it looks quite technical, I will give here for the sake of quick reference the definition of

$C=RHom(R\Gamma_{ét}(Spec(\mathcal{O_{k_\infty}}[\frac{1}{\Sigma}]),\mathbb{Q}_p/\mathbb{Z}_p),\mathbb{Q}_p/\mathbb{Z}_p)$.

This $C$ is strongly correlated to $X$, namely $H^0(C)=\mathbb{Z}_p$ and $H^{-1}(C)=X$.

Now my question: An expert in the field told me that this $\mathbb{Z}_p$ is 'moraly' related to the pole of a zetafunktion. How is this?

Is this even related to the Main Conjecture, where evalutations at representations of $G$ and $p$-adic interpolation play the lead role? As far as I understand it, the trivial representation, leading to the zeta function, is left undealt with.

I apologize for my ignorance on this basic question of the field.

For all that follows, $p$ is a fixed odd prime. In the formulation of the Noncommutative Main Conjecture of Iwasawa theory one uses étale cohomology to define an algebraic object analogous to Iwasawa's 'characteristic ideal' in $\Lambda(G)$ for $G=Gal(k_\infty/k)$, with $k^{cyc}\subset k_\infty$ and $\mu=0$:

Let $M_\Sigma$ be the maximal abelian, pro-$p$, outside of $\Sigma$ unramified extension of $k_\infty$ for a finite set of primes of $k$, $\Sigma$. Then $X:=Gal(M_\Sigma/k_\infty)$ is a $\Lambda(G)$ module. If $G$ has elements of order $p$ we are prevented from seeing this $X$ a relative $K_0$ group, associated to a denominator set, $S$, of $\Lambda(G)$.

Now étale cohomology enters the picture: We use it to define a complex $C$ of $\Lambda(G)$-modules which is $S$-acyclic and quasi-isomorphic to a bounded complex of finitely generated $\Lambda(G)$-modules. Although it looks quite technical, I will give here for the sake of quick reference the definition of

$C=RHom(R\Gamma_{ét}(Spec(\mathcal{O_{k_\infty}}[\frac{1}{\Sigma}]),\mathbb{Q}_p/\mathbb{Z}_p),\mathbb{Q}_p/\mathbb{Z}_p)$.

This $C$ is strongly correlated to $X$, namely $H^0(C)=\mathbb{Z}_p$ and $H^{-1}(C)=X$.

Now my question: An expert in the field told me that this $\mathbb{Z}_p$ is 'morally' related to the pole of a zeta function. How is this?

Is this even related to the Main Conjecture, where evalutations at representations of $G$ and $p$-adic interpolation play the lead role? As far as I understand it, the trivial representation, leading to the zeta function, is left undealt with.

I apologize for my ignorance on this basic question of the field.