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Frank Thorne
Frank Thorne
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I saw the following nice formula in an unpublished paper of H. Cohen.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

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

between the characters of $S_4$ induced from the trivial characters of the subgroups of $Gal(\widetilde{L}/\mathbb{Q})$$\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?

I saw the following nice formula in an unpublished paper of H. Cohen.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 $Gal(\widetilde{L}/\mathbb{Q})$ 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?

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?

Source Link
Frank Thorne
Frank Thorne
  • 7.3k
  • 9
  • 62
  • 78

Generalizations/applications of a formula for the Dedekind zeta function?

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 $Gal(\widetilde{L}/\mathbb{Q})$ 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?