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A naive and idle number theory question from a topologist (but not a knot theorist):

I have heard it said (and this came up recently at MO) that there is a fruitful analogy between Spec $\mathbb Z$ and the $3$-sphere. I gather that from an etale point of view the former is $3$-dimensional and simply connected; from the same point of view the subschemes Spec $\mathbb Z/p$ are $1$-dimensional and very much like circles; and the Legendre symbols for two odd primes that figure in quadratic reciprocity are said to be analogous to linking numbers of knots. So, prompted by a recent MO question, I started thinking:

The abelianized fundamental group of the complement of Spec $\mathbb Z/p$ (the group of $p$-adic units) is not terribly different from the abelianized fundamental group of a knot complement (an infinite cyclic group). For nontrivial knots, there is a lot more to the fundamental group of a knot complement than its abelianization. The next little bit, the abelianization of the commutator subgroup (or $H_1$ of the infinite cyclic cover) has an action of that infinite cyclic group, and I recall that the Alexander polynomial of the knot may be created out of this action.

So there must be some analogue of that in number theory, right? Like, some construction involving ideal class groups or idele class groups of $p$-power cyclotomic fields can be interpreted as the Alexander polynomial of a prime number?

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Short answer: The group of ideles is the group of units of the ring of adeles, so the group of ideles is a multiplicative thing and the group of adeles is an additive thing. The ideles map onto the ideals. I am guessing that "adeles" was coined after "ideles", with "ad-" indicating "additive". Who wants to tell the true story? –  Tom Goodwillie Jul 10 '10 at 0:37
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Tom, your guess is basically correct, but I heard that Weil (who coined the name "adeles" to replace the previous "valuation vectors") also intended it as a kind of joke, since adele is a French girl's name too. Then it stuck. –  BCnrd Jul 10 '10 at 1:45
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Tom, ideles were invented by Chevalley around 1940 for his non-analytic development of class field theory (all priori work required analytic input at some point), and I believe he gave them the name "ideles" because of how they were used to unify generalized ideal class groups into quotients of a single structure. However, due to the application in class field theory, he gave them a topology with far fewer open sets than the one now used. That they were units of a ring was only recognized later (presumably quite soon), since the context for their invention had no need for it. –  BCnrd Jul 10 '10 at 3:48
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I would have given a substantial answer to this question if I were not so lazy. Instead, let me point to the papers: Coates, John; Fukaya, Takako; Kato, Kazuya; Sujatha, Ramdorai; Venjakob, Otmar The $\rm GL_2$ main conjecture for elliptic curves without complex multiplication. Publ. Math. Inst. Hautes Études Sci. No. 101 (2005), 163--208; Fukaya, Takako; Kato, Kazuya A formulation of conjectures on $p$-adic zeta functions in noncommutative Iwasawa theory. Proceedings of the St. Petersburg Mathematical Society. Vol. XII, 1--85, Amer. Math. Soc. Transl. Ser. 2, 219; cont. –  Minhyong Kim Jul 11 '10 at 6:15
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mathematik.uni-regensburg.de/preprints/Forschergruppe/… These illustrate how homotopy-theoretic the incarnations have become. In brief, the current view is that the Iwasawa polynomial=p-adic L-function should be viewed as a path in K-theory space. –  Minhyong Kim Jul 11 '10 at 6:18

2 Answers 2

up vote 19 down vote accepted

This article appears to discuss the relationship between Alexander polynomials in knot theory and Iwasawa polynomials in number theory, although I haven't looked at it in detail. I discovered this paper in This Week's Finds 257, which gives a number of references for this sort of thing.

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Yep, the Iwasawa polynomial (which, just as you say, records the behavior of the ideal class groups of p-power cyclotomic fields) is analogous to the Alexander polynomial. –  JSE Jul 10 '10 at 2:38

A whole book has recently appeared (Primes and Knots, Edited by Toshitake Kohno, University of Tokyo, Japan, and Masanori Morishita, Kyushu University, Fukuoka, Japan).

This volume deals systematically with connections between algebraic number theory and low-dimensional topology. Of particular note are various inspiring interactions between number theory and low-dimensional topology discussed in most papers in this volume. For example, quite interesting are the use of arithmetic methods in knot theory and the use of topological methods in Galois theory. Also, expository papers in both number theory and topology included in the volume can help a wide group of readers to understand both fields as well as the interesting analogies and relations that bring them together.

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