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In a comment on Tom Goodwillie's question about relating the Alexander polynomial and the Iwasawa polynomial, Minhyong Kim makes the cryptic but tantalizing statement:

In brief, the current view is that the Iwasawa polynomial=p-adic L-function should be viewed as a path in K-theory space.

As somebody who has been trying, for a long time, to understand the Alexander polynomial, and who believes there is something "more basic" underlying its appearances in mathematics in various guises (my particular interest has to do with why it turns up as the wheels part of the Kontsevich invariant, as per Melvin-Morton-Rozansky), I'd really like to understand this comment.

Question: What is "K-theory space" supposed to be, and why should a path in it be a natural object to be examining? And why should we expect such a thing to give equivalent data to the Iwasawa polynomial/ Alexander polynomial?
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Here's a stab in the dark, after glancing at the paper Minhyong gives a link to. The algebraic $K$-theory groups of a ring $R$ are the homotopy groups of a space $K(R)$, so $K_q(R)=\pi_q K(R)$. So his "$K$-theory space" is $K(R)$ for some suitable $R$. If a path in K-space is actually a closed path, i.e., a loop, then it gives an element in $\pi_1K(R)=K_1(R)$. So my guess is that we are supposed to be able to think of Iwasawa polynomial/p-adic L-functions, at least in very nice cases, as certain elements in $K_1(R)$ for some suitable $R$. (continued ...) –  Charles Rezk Sep 1 '10 at 20:09
... Looking at Witte's paper that Kim links to, it looks like we are supposed to take $R=\mathbb{Z}_\ell[[G]]$ for $G$ related to the Galois group; thus, $R$ could be the Iwasawa ring, or maybe some (possibly non-commutative) analogue of it. Actually, that's a pretty cool picture! (If I'm right in how I'm reading reading it.) Don't know what that has to do with the Alexander polynomial ... –  Charles Rezk Sep 1 '10 at 20:14
Well, I guess I know that Alexander polynomial has something to do with Reidemeister torsion, which has something to do with Whitehead torsion, which has something to do with $K_1(\pi_1M)$. But now I'm out of my league ... –  Charles Rezk Sep 1 '10 at 20:22
What Charles writes is correct. The paper of Coates et. al. gives a good deal of motivation. I'll post an answer in about a week if no one else is forthcoming before that. I'm somewhat preoccupied at the moment. –  Minhyong Kim Sep 1 '10 at 22:14
0 Perhaps I'll add two comments for now: In the K1 space of a possibly non-commutative Iwasawa algebra, it really is a path. It only becomes a loop, i.e., an element of $K_1$ after localizing suitably. This is a rather interesting phenomenon. But the work of Coates, Kato, and so on concentrate on $K_1$-space. And then, Witte proposes that one should lift the path to the whole $K$-theory space. That way, one can pay closer attention to homotopies of paths –  Minhyong Kim Sep 2 '10 at 2:10
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1 Answer

up vote 15 down vote accepted

I know nothing about Alexander polynomials but let me try to answer the Iwasawa theory part. As is well known, in classical Iwasawa theory one considers cyclotomic $\mathbb{Z}_p$ extension $F{\infty}$ of $F$. We take the $p$-part of the ideal class group $A_n$ of the intermediate extension $F_n$ of $F$ of degree $p^n$. The inverse limit of $A_n$ with respect to norm maps, say $A$, has an action of $G= Gal(F_{\infty}/F)$. Since $A$ is pro-$p$, it becomes a $\mathbb{Z}_p[G]$-module. However, it is not finitely generated over this group ring (and for various other reasons) one considers the completion $\mathbb{Z}_p[[G]]$ of $\mathbb{Z}_p[G]$. Since $A$ is compact, it becomes a $\mathbb{Z}_p[[G]]$-module. As $G \cong \mathbb{Z}_p$, the ring $\mathbb{Z}_p[[G]] \cong \mathbb{Z}_p[[T]]$, the power series ring in variable $T$. There is a nice structure theory for finitely generated modules over $\mathbb{Z}_p[[T]]$. The module $A$ is a torsion $\mathbb{Z}_p[[G]]$-module (i.e.$Frac(\mathbb{Z}_p[[G]]) \otimes A = 0$). For such modules one can define the characteristic ideal using the structure theory. Iwasawa's main conjecture asserts that there is a canonical generator for this ideal called the $p$-adic $L$-function.

In generalised Iwasawa theory (more precisely, to formulate the generalised main conjecture à la Kato), one wants to consider extensions whose Galois groups are not necessarily $\mathbb{Z}_p$ (but most formulations of the main conjecture still require that the cyclotomic $\mathbb{Z}_p$-extension of the base field be in the extension). For the completed $p$-adic groups rings of such Galois groups, the structure theory completely breaks down even if the Galois group is abelian.

However, one can still show that $A$ is a torsion Iwasawa module (which again just means that $Frac(\mathbb{Z}_p[[G]]) \otimes A = 0$. Note that it is always possible to invert all non-zero divisors in a ring even in the non-commutative setting). Hence the class of $A$ in the group $K_0(\mathbb{Z}_p[[G]])$ is zero. Strictly speaking, here I must assume that $G$ has no $p$-torsion so that I can take a finite projective resolution of $A$, or I must work with complexes whose cohomologies are closely related to $A$. But I will sinfully ignore this technicality here. Now, since the class of $A$ in $K_0(\mathbb{Z}_p[[G]])$ is zero, there is a path from $A$ to the trivial module 0 in the $K$-theory space. In Iwasawa theory this is most commonly written as

There exists an isomorphism $Det_{\mathbb{Z}_p[[G]]}(A)$ $\to$ $Det(0)$.

This isomorphism replaces the characteristic ideal used in the classical Iwasawa theory. The $p$-adic $L$-function then is a special isomorphism of this kind. (Well one has to be careful about the uniqueness statement in the noncommutative setting but it is a reasonably canonical isomorphism). Hence the main conjecture now just asserts existence of such a $p$-adic $L$-function.

Thus the $p$-adic $L$-function may be thought of as a canonical path in the $K$-theory space joining the image of Selmer module (or better- a Selmer complex), such as the ideal class group in the above example, and the image of the trivial module.

I hope this answer helps until Minhyong sheds more light on his remarks and relations between $p$-adic $L$-functions and the Alexander polynomials.

[EDIT: Sep. 30th] To answer Daniel's questions below- 1) Take the projective resolution of A to define its class in $K_0$. In any case the $K$-theory of the category of finitely generated modules is the same as the $K$-theory of finite generated projective modules. 2) I do not know if there are any sensible/canonical multiplicative sets at which to localise $\mathbb{Z}_p[G]$. The modules considered in Iwasawa theory are usually compact (or co-compact depending on whether you take inverse limit with respect to norms or direct limit with respect to inclusion of fields in a tower) and so the action of the ring $\mathbb{Z}_p[G]$ extends to an action of the completion $\mathbb{Z}_p[[G]]$ by which we mean $\varprojlim \mathbb{Z}_p[G/U]$, where $U$ runs through open normal subgroups of $G$. 3) We do not usually get a loop in the $K$-theory space of $\mathbb{Z}_p[[G]]$. However, modules which come up from arithmetic are usually torsion as $\mathbb{Z}_p[[G]]$-modules i.e. every element in the module is annihilated by a non-zero divisor or in other words if $X$ is the module then $Frac(\mathbb{Z}_p[[G]]) \otimes X=0$. (it is usually very hard to prove that the modules arising are in fact torsion). But once we know that they are torsion we get a loop in the $K$-theory space of $Frac(\mathbb{Z}_p[[G]])$. If we know that for some multiplicatively closed subset $S$ of $\mathbb{Z}_p[[T]]$ annihilates $X$ i.e. $(\mathbb{Z}_p[[G]]_S) \otimes X=0$, then we already get a loop in the $K$-theory space of $\mathbb{Z}_p[[G]]$. In noncommutative Iwasawa theory it is often necessary to work with a multiplicative set strictly smaller than all non-zero divisors.

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sorry- I tried to improve formatting, and ended up ruining it... help? –  Daniel Moskovich Sep 3 '10 at 2:17
As you see, I am new here and had some trouble getting it into the format I would have liked. For now I just rolled it back to the original. –  Mahesh Kakde Sep 3 '10 at 2:36
+6, if that had been possible. –  Chandan Singh Dalawat Sep 8 '10 at 10:24
I have just (finally) read through this carefully, and it's tremendously helpful! A few questions: 1) If A isn't projective, what is the definition of K_0? 2) Why do we need to localize all the way to Z_p[[G]]? Is the Cohn localization of Z_p[G] not enough to ensure finite generation? 3) How can you get a closed path? The path constructed above doesn't look like it can ever be closed for A non-zero. –  Daniel Moskovich Sep 28 '10 at 17:46
I'm sorry I haven't written anything thus far. But I can see the question has been taken up by Mahesh, who has knowledge of the subject infinitely superior to mine. I hope people don't mind if I advertise that Mahesh has among the deepest results thus far on the proof of the non-abelian main conjectures. The little I know I learned from him. –  Minhyong Kim Sep 30 '10 at 11:24
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