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I saw the following nice formula in an unpublished paper of H. Cohen. Let $L$ be a quartic field, whose Galois closure $\widetilde{L}$ has Galois group $S_4$. Denote the cubic resolvent field of $L$ by $K_3$, and let $K_6$ be the Galois closure of $K_3$.

Then, we have the following relation among Dedekind zeta functions:

$$\zeta_L(s) = \frac{ \zeta(s) \zeta_{K_6}(s) }{\zeta_{K_3} (s)}.$$

This is proved using the formalism of Artin $L$-functions, and reflects the relation

$$ Ind_{G_L} 1 = 1 + Ind_{G_{K_6}} 1 - Ind_{G_{K_3}} 1$$

between the characters of $S_4$ induced from the trivial characters of the subgroups of $\mathrm{Gal}(\widetilde{L}/\mathbb{Q})$ fixing $G_L$, $G_{K_6}$, $G_{K_3}$ respectively. So essentially this is a calculation in group representation theory.

One nice consequence is the simple relation $D_L = D_{K_6} / D_{K_3}$ between the discriminants of these number fields. (This may be proved by comparing the conductors in the functional equation.)

I imagine this is old hat, but this formula really struck me. Two questions:

(1) Are there other similarly nice relations between other families of Dedekind zeta functions?

(2) Have this or related formulas seen other interesting applications in number theory?

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See the end of Frohlich and Taylor's algebraic number theory book for more examples in this direction (I think it is what they call Brauer relations). I used some similar relations for the zeta-function of a cubic field and its Galois closure to do calculations in some handouts at math.uconn.edu/~kconrad/blurbs called "Invariants of the splitting field of a cubic I, II,...,V". –  KConrad Mar 30 '12 at 3:24
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@KConrad: These notes are a gold mine! I will definitely look at many of them when my turn comes to teach algebraic number theory. –  Frank Thorne Mar 30 '12 at 13:46
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1 Answer

up vote 8 down vote accepted

As Keith says, such relations between permutation representations are often called Brauer relations, because Brauer was the first one to note that such isomorphisms of permutation representations give rise to relations between zeta functions (Kuroda noticed the same phenomenon at the same time). Any non-cyclic group has Brauer relations, so in any $G$-extension of number fields with $G$ non-cyclic, you will get relations between zeta functions of the intermediate fields. Moreover, the space of Brauer relations is a finitely generated abelian group, and its rank is equal to the number of conjugacy classes of non-cyclic subgroups of $G$ (this is a consequence of Artin's induction theorem, together with the fact that the number of irreducible rational representations of $G$ is equal to the number of conjugacy classes of cyclic subgroups). So that's precisely the number of essentially different relations between zeta functions that you get this way in any given Galois extension of number fields. The Brauer relations in an arbitrary finite group are completely classified in this joint work of mine with Tim Dokchitser.

Such relations have lots of applications. For example numerous papers by Bart de Smit, Perlis, and others investigated relations that only consist of two groups. See my answer here for a description of some of their results. In this paper I used these relations to investigate integral units as Galois modules, and in this paper Bart and I generalised those findings.

In that paper with Bart we make use of the fact that you also get similar relations between other $L$-functions, e.g. $L$-functions of elliptic curves. This had been used to great effect by Tim and Vladimir Dokchitser (see here and here) to obtain results in the direction of the Birch and Swinnerton-Dyer conjecture. Also, Bley and Boltje have used Brauer relations to obtain relations between $p$-parts of class number, of Tate-Shafarevich groups, and of other interesting number theoretic invariants.

A final remark: to get the relations between discriminants from Brauer relations, you do not need zeta functions. They follow immediately from the conductor-discriminant formula.

The references to the original papers by Brauer and Kuroda are:

R. Brauer, Beziehungen zwischen Klassenzahlen von Teilkörpern eines Galoisschen Körpers, Math. Nachr. 4 (1951), 158–174 and

S. Kuroda, Über die Klassenzahlen algebraischer Zahlkörper, Nagoya Math. J. 1 (1950), 1–10.

If you search for forward citations, you will find lots of literature.

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What a great answer! Thank you. –  Frank Thorne Mar 30 '12 at 13:35
    
One question about your "final remark": My impression of the proofs is that the proof of the relations among Artin $L$-functions is quite similar to that of the conductor-discriminant formula (at least, the proofs given in Neukirch). Although the c-d formula can be proved without Artin $L$-functions, these $L$-function relations seem to be the biggest, most useful hammer in the toolbox. However I am not an expert in the subject (and evidently you are!) If you think my perspective is at all misleading, would you please explain? Thank you very much! –  Frank Thorne Mar 30 '12 at 13:42
    
Thank you, Frank, for the kind words! Actually, you are quite right: the proof of the conductor-discriminant formula is the same so-called Artin formalism. –  Alex B. Mar 30 '12 at 13:56
    
@Alex: This is really a great answer! –  Guillermo Mantilla Apr 12 '12 at 9:44
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