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.



Characteristic Complexes in Iwasawa theory
The reason why one cannot take the class of $X$ in the relative $K\_0$ when $G$ has $p$-torsion is because $X$ may not have finite resolution by finitely generated projective $\Lambda(G)$-modules. This is necessary even if $G$ is abelian. Hence we take the complex $C$ above. $\mathbb{Z}\_p$ appearing there is indeed interpreted as a pole. One way thinking about this in the commutative Iwasawa theory is as follows- if $G$ is of the form $H \times \mathbb{Z}\_p$, with $H$ a finite abelian group, then for each one dimensional character $\chi$ of $H$ there is a $p$-adic $L$-function, say $L\_p(\chi)$, constructed by Deligne-Ribet, Cassou-Nogues, Barsky. $L\_p(\chi)$ is expected to have a simple pole only when $\chi$ is the trivial character. On the algebraic side one has to make sense of what a characteristic ideal for $X$ means when order of $H$ is divisible by $p$. One can either use the $K$-theory formulation as suggested by Fukaya-Kato or one can do something more directly but only get weaker information. Look at $V:=X \otimes \overline{\mathbb{Q}}\_p$. This is a finite dimensional $\overline{\mathbb{Q}}\_p$ vector space. For each $\chi$ we can consider $V^{(\chi)}$, the $\chi$ isotypic part. Let $f(\chi)$ be the characteristic polynomial of $1-\gamma$ acting on $V$ (Here $\gamma$ is a fixed topological generator of $\mathbb{Z}\_p$). Then for every non-trivial $\chi$ the polynomial $f(\chi)$ generates the same ideal as $L\_p(\chi)$ in the ring $\mathbb{Z}\_p[\chi][[\mathbb{Z}\_p]] = \mathbb{Z}\_p[\chi][[T]]$. However, when $chi$ is the trivial character, we have that $f(\chi)$ and $TL\_p(\chi)$ generate the same ideal. Or $f(\chi)/T$ is essentially same as the $p$-adic $L$-function. So apart from the information about the characteristic element of $X$, the $p$-adic $L$-function has this additional pole (the $T$ in the denominator) corresponding to the module $\mathbb{Z}\_p$ with the trivial action. This is exactly what happens in the noncommutative situation as well. It is not a new phenomenon of the noncommutative theory- it was always there. 


The answer to your next question is yes, it is related to the main conjecture. We conjecture that there is an element $\zeta \in K\_1(\Lambda(G)\_S)$ which maps to the class of $C$ in $K\_0(\Lambda(G), \Lambda(G)\_S)$ and which is related to values of $L$-functions at Artin representations of $G$ at odd negative integers i.e. evaluations at even positive Tate twists of Artin representations of $G$. We now have this $\zeta$ and the main conjecture in the noncommutative situation and again one expects $\zeta$ a simple pole when evaluated at the trivial representation of $G$. 


