Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question

1 Answer 1

up vote 5 down vote accepted

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 $G$.

share|improve this answer
    
Thank you very much for this! I understand the tensor product in the definition of V is over Z_p and for V being of finite dimension you have to use the Theorem of Ferrero-Washington? –  Konrad Sep 30 '10 at 19:51
    
Yes, the tensor is over $\mathbb{Z}_p$. I wanted to put it but somehow whenever I tried doing that I did not get the desired typesetting output. The fact that $V$ is finite dimensional does not use the theorem of Ferrero-Washington about vanishing of $\mu$ invariant (which is anyways available only over abelian extensions of $\mathbb{Q}$). It follows from the fact that $X$ is torsion $\Lambda(G)$-module which was proved by Iwasawa. –  Mahesh Kakde Oct 1 '10 at 10:36

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.