Characteristic Complexes in Iwasawa theory

Question

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.

Answer 1

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 $G$.