An analogue of Kummer's relation between the ideal class number and the unit index of cyclotomic fields

TL;DR

This paper derives a new class number formula for maximal real subfields of cyclotomic fields, introduces a novel type of cyclotomic units, and establishes a relation akin to Kummer's between class number and unit index.

Contribution

It provides a new class number formula for real subfields of cyclotomic fields and constructs new cyclotomic units, extending Kummer's relation.

Findings

Derived a formula for the special value of Euler-Dirichlet L-function at s=1

Constructed a new type of cyclotomic units in $ ext{Q}( ext{μ}_{p^{n}})$

Established a relation between class number and unit index similar to Kummer's

Abstract

In this paper, we obtain a formula for the special value of Euler-Dirichlet $L$-function $L_E(s,\chi)$ at $s=1$. This leads to another class number formula of $\mathbb{Q}(\mu_{m})^{+}$, the maximal real subfield of $m$th cyclotomic field. From this formula, we construct a new type of cyclotomic units in $\mathbb{Q}(\mu_{p^{n}})$, which implies a similar Kummer's relation between the ideal class number of $\mathbb{Q}(\mu_{p^{n}})^{+}$ and the unit index.

Equations

$$L_E(s,\chi )=\sum _{n=1}^{\infty }\frac{(-1{)}^{n-1}\chi (n)}{n^s},\text{Re}(s)>0,$$

$$\eta (s)=\sum _{n=1}^{\infty }\frac{(-1{)}^{n-1}}{n^s},\text{Re}(s)>0.$$

$$L(s,\chi )=\sum _{n=1}^{\infty }\frac{\chi (n)}{n^s},$$

$${\zeta }_f=e^{\frac{2\pi i}{f}}=\cos \left(\frac{2\pi }{f}\right)+i\sin \left(\frac{2\pi }{f}\right)$$

$$\tau (\chi )=\sum _{r(\text{mod}f)}\chi (r){\zeta }_f^r,$$

$$L(1,\chi )$$

$$=-\frac{\tau (\chi )}{f}\sum _{k=1}^f\overline{\chi }(k)\log |1-{\zeta }_f^k|,$$

$$L(1,\chi )$$

$$\frac{2^{r_1}(2\pi {)}^{r_2}hR}{w{d}^{1/2}}=\prod _{\chi \ne 1}L(1,\chi ),$$

$$w$$

$$h$$

$$R$$

$$d$$

$$G=\mathbb{Z}(m{)}^{\ast }/\pm 1\text{and}{G}_{\chi }=\mathbb{Z}(f_{\chi }{)}^{\ast }/\pm 1$$

$$h^{+}=\frac{1}{R^{+}}\prod _{\chi \ne 1}\sum _{k\in G_{\chi }}-\chi (k)\log |1-{\zeta }_{f_{\chi }}^k|,$$

$$g_k=\frac{{\zeta }_m^k-1}{{\zeta }_m-1}.$$

$${\zeta }_m^{-v}{g}_k\text{for}v=\frac{k-1}{2}$$

$$E/\mathcal{E}\cong E^{+}/{\mathcal{E}}^{+}.$$

$$h^{+}=(E:\mathcal{E})=(E^{+}:{\mathcal{E}}^{+}).$$

$$L_E(1,\chi )$$

$$=\frac{\tau (\chi )}{f}\sum _{k=1}^f\overline{\chi }(k)\log |1+{\zeta }_f^k|,$$

$$\text{if}\chi \text{ is even},\text{i.e.}\chi (-1)=1.$$

$$L_E(1,\chi )$$

$$\text{if}\chi \text{ is odd},\text{i.e.}\chi (-1)=-1.$$

$$\eta =\prod _{\chi \ne 1}(1-\chi (2))$$

$$\eta h^{+}=\frac{1}{R^{+}}\prod _{\chi \ne 1}\sum _{k\in G_{\chi }}\chi (k)\log |1+{\zeta }_{f_{\chi }}^k|,$$

$${\tilde{g}}_k=\frac{{\zeta }_m^k+1}{{\zeta }_m+1}$$

$${\zeta }_m^{-v}{\tilde{g}}_k\text{for}v=\frac{k-1}{2}$$

$$E/\tilde{\mathcal{E}}\cong E^{+}/{\tilde{\mathcal{E}}}^{+}.$$

$$|\eta |h^{+}=(E:\tilde{\mathcal{E}})=(E^{+}:{\tilde{\mathcal{E}}}^{+}).$$

$$n=qf+r,\text{where 1≤r≤f and q=0,1,2,…,}$$

$$-L_E(s,\chi )$$

$$=\sum _{(r,f)=1}\chi (r)\sum _{n\equiv r(\text{mod}f)}\frac{(-1{)}^n}{n^s},$$

$$\sum _{n=1}^{\infty }\frac{(-1{)}^n{c}_n}{n^s},$$

$$c_n=\{\begin{array}{ll}1& \text{for }n\equiv r(\mathrm{mod}f)\\ 0& \text{for }n\not\equiv r(\mathrm{mod}f).\end{array}$$

Full text

*Corresponding author

1. Introduction

Throughout the paper, $\chi$ will always be a primitive Dirichlet character with conductor $f_{\chi}$ or $f$. Therefore, for $n\in\mathbb{Z}$, $\chi(n)=0\Leftrightarrow\textrm{gcd}(n,f)>1$.

An alternating form of Dirichlet $L$-function:

[TABLE]

is called Euler-Dirichlet $L$-function. We see that $L_{E}(s,\chi)$ (1.1) is indeed the following Dirichlet eta function with a character:

[TABLE]

The above $\eta(s)$ is a particular case of Witten’s zeta functions in mathematical physics (see [5, p. 248, (3.14)]) and it has been used by Euler to obtain a functional equation of Riemann zeta function $\zeta(s)$ (see [7, p.273–276]). There is also a connection between $L_{E}(s,\chi)$ and the ideal class group of the $p^{n+1}$-th cyclotomic field where $p$ is a prime number. For details, we refer to a recent paper [3], especially [3, Propositions 3.2 and 3.4].

In this paper, we obtain a formula for the special value of Euler-Dirichlet $L$-function $L_{E}(s,\chi)$ at $s=1$ (see Theorem 1.5). As a consequence, we get another class number formula of $\mathbb{Q}(\mu_{m})^{+}$, the maximal real subfield of $m$th cyclotomic field (see Theorem 1.6). From this formula, we construct a new type of cyclotomic units in $\mathbb{Q}(\mu_{p^{n}})$ (see Eq. (1.13)), which implies a similar Kummer’s relation between the class number of $\mathbb{Q}(\mu_{p^{n}})^{+}$ and the unit index (see Theorem 1.8).

1.1. Background

The Dirichlet $L$-function $L(s,\chi)$ ($L$-series) is defined by

[TABLE]

where the series on the right is absolutely convergent for Re$(s)>1$ and is conditionally convergent for Re$(s)>0$ for non-principal $\chi$ (see [2]). Let

[TABLE]

be a primitive $f$th root of unity and

[TABLE]

be the Gauss sum corresponding to the character $\chi$, where $r$ runs through a full (or a reduced) system of residues modulo $f.$ If $\chi$ is non-principal, the following result on the special value of $L(s,\chi)$ at $s=1$ is well-known.

Theorem 1.1** (See Washington [6, p.38, Theorem 4.9]).**

[TABLE]

This formula has an important application in algebraic number theory by connecting $L(1,\chi)$ with the class number formula of abelian fields $K/\mathbb{Q}$, that is:

Theorem 1.2** (See Lang [4, p.77]).**

[TABLE]

where the product is taken over all the primitive characters induced by the characters of Gal($K/\mathbb{Q}$) and

[TABLE]

If $K$ is real, then $r_{1}=[K:\mathbb{Q}]$ and $r_{2}=0$; if $K$ is complex, then $r_{1}=0$ and $r_{2}=\frac{1}{2}[K:\mathbb{Q}]$.

Assume that $m$ is odd or $m\equiv 0~{}(\textrm{mod}~{}4)$, $K=\mathbb{Q}(\mu_{m})$ and $K^{+}=\mathbb{Q}(\mu_{m})^{+}$ be the $m$-th cyclotomic field and its maximal real subfield, respectively; $h$ and $h^{+}$ be the class number of $K$ and $K^{+}$, respectively.

For convenience of the notations, we also denote $\mathbb{Z}/m\mathbb{Z}$ by $\mathbb{Z}(m)$. Let $f_{\chi}$ be the conductor of $\chi$. Introduce the group

[TABLE]

Combining (1.6) and (1.7), we have the class number formula of $K^{+}$.

Theorem 1.3** (See Lang [4, p.81]).**

[TABLE]

where the product over $\chi\neq 1$ is taken over the non-trivial characters of $G$, or equivalently, the non-trivial even characters of $\mathbb{Z}(m)^{*}$.

For $k$ prime to $m$, we let

[TABLE]

Then $g_{k}$ is called a cyclotomic unit. It is easy to see that $g_{k}$ is equal to a real unit times a root of unity. Since $\zeta_{m}^{k}$ only depends on the residue class of $k$ mod $m$, without loss of generality, we may assume that $k$ is odd. Then

[TABLE]

is real (i.e. fixed under $\sigma_{-1}$), and call it the real cyclotomic unit.

Let $E$ be the group of unit in $K$ and $\mathcal{E}$ be the the subgroup of $E$ generated by the roots of unity and the cyclotomic units. Let $E^{+}$ be the group of unit in $K^{+}$ and $\mathcal{E}^{+}$ be the the subgroup of $E^{+}$ generated by $\pm 1$ and real cyclotomic units. Assume $m=p^{n}$ is a prime power, we have

[TABLE]

From (1.9) and the Dedekind determinant formula (see Theorem 5.1), we have the following Kummer’s result related to the class number $h^{+}$ and the unit indexes $(E:\mathcal{E})=(E^{+}:\mathcal{E}^{+})$.

Theorem 1.4** (See Lang [4, p.85, Theorem 5.1]).**

Let $K=\mathbb{Q}(\mu_{m})$, $K^{+}=\mathbb{Q}(\mu_{m})^{+}$ and $h^{+}$ be the class number of $K^{+}$. Assume $m=p^{n}$ is a prime power. Then

[TABLE]

1.2. Our results

First, we will calculate the special value of $L_{E}(s,\chi)$ at $s=1$ (comparing with Theorem 1.1 above).

Theorem 1.5**.**

[TABLE]

Assume that $m$ is odd or $m\equiv 0~{}(\textrm{mod}~{}4)$, $K=\mathbb{Q}(\mu_{m})$ and $K^{+}=\mathbb{Q}(\mu_{m})^{+}$ be the $m$-th cyclotomic field and its maximal real subfield, respectively; $h$ and $h^{+}$ be the class number of $K$ and $K^{+}$, respectively. Denote by

[TABLE]

where the product over $\chi\neq 1$ is taken over the non-trivial characters of $G$, or equivalently, the non-trivial even characters $\mathbb{Z}(m)^{*}$. Combing (1.10) and the class number formula of abelian fields (see (1.7)), we also have another class number formula of $K^{+}$ (comparing with Theorem 1.3 above).

Theorem 1.6**.**

[TABLE]

where the product over $\chi\neq 1$ is taken over the non-trivial characters of $G$, or equivalently, the non-trivial even characters $\mathbb{Z}(m)^{*}$.

For $k$ prime to $m$, we define a new type of cyclotomic units to be

[TABLE]

(comparing with the definition of $g_{k}$ above). Then $\tilde{g}_{k}$ is equal to a real unit times a root of unity. Since $\zeta_{m}^{k}$ only depends on the residue class of $k$ mod $m$, without loss of generality, we may assume that $k$ is odd. Then

[TABLE]

is real (i.e. fixed under $\sigma_{-1}$), and we call it a new type of cyclotomic units.

Let $E$ be the group of unit in $K$ and $\tilde{\mathcal{E}}$ be the subgroup of $E$ generated by the roots of unity and the new type cyclotomic units defined above. Let $E^{+}$ be the group of unit in $K^{+}$ and $\tilde{\mathcal{E}}^{+}$ be the the subgroup of $E^{+}$ generated by $\pm 1$ and the new type real cyclotomic units introduced above. We have the following isomorphism.

Proposition 1.7**.**

Assume $m=p^{n}$ is a prime power, we have

[TABLE]

From (1.12) and the Dedekind determinant formula (see Theorem 5.1), we also have a new formula related to $h^{+}$ and the unit indexes $(E:\tilde{\mathcal{E}})$ and $(E^{+}:\tilde{\mathcal{E}}^{+})$ (comparing with Theorem 1.4 above).

Theorem 1.8**.**

Let $K=\mathbb{Q}(\mu_{m})$, $K^{+}=\mathbb{Q}(\mu_{m})^{+}$ and $h^{+}$ be the class number of $K^{+}$. Assume $m$ is a prime power. If either $m=2^{n}$; or $m=p^{n}$ is an odd prime power and $-1,2$ generate the group $\mathbb{Z}(m)^{*}$, then $\eta\neq 0$ and

[TABLE]

*Otherwise $\eta=0$ and $\mathcal{E}$ (resp. $\mathcal{E}^{+}$) is of infinite index in $E$ (resp. $E^{+}$). *

2. Proof of Theorem 1.5

Let $\chi$ be a non-trivial Dirichlet character with conductor $f$. We rearrange the terms in the series for $L_{E}(s,\chi)$ according to the residue classes mod $f.$ That is, we write

[TABLE]

and obtain

[TABLE]

since $\chi(r)=0$ if $(r,f)>1.$ The inner series can be written in the form

[TABLE]

where

[TABLE]

To find a convenient way of writing the coefficients $c_{n},$ we consider the following formula:

[TABLE]

where

[TABLE]

is a primitive $f$th root of unity. We remark

[TABLE]

(see [1, p. 332]). Therefore, combing with (2.1), (2.2) and (2.4) we have the identity

[TABLE]

since $\sum_{(r,f)=1}\chi(r)=0$ if $\chi$ is not the principal character.

The Gauss sum $\tau(\chi)$ (1.5) depends not only on $\chi$, but also on the choice of the root $\zeta_{f}.$ Throughout the proof of this theorem, we always assume $\zeta_{f}$ is $\cos(2\pi/f)+i\sin(2\pi/f).$

We see that [6, Lemma 4.7]

[TABLE]

Using (2.6) we can write (2.5) in the form

[TABLE]

The above series (2.7) converges for $0<s<\infty$ and represents a continuous function of $s.$ Hence we may set $s=1$ in this last equation and obtain

[TABLE]

To find the sum of the inner series on the right in (2.8), we consider

[TABLE]

It is well known that the analytic function $\log(1+z)$ defined by (2.9) has $z=-1$ as its only singular point at finite distance. Since this series also converges at the point $z=\zeta_{f}^{-k}$ (on the unit circle), then by Abel’s theorem, we have

[TABLE]

and hence

[TABLE]

thus we have obtained a finite expression for the series $L_{E}(1,\chi).$

The formula (2.11) can be further investigated and considerably simplified as follows. Let

[TABLE]

where $k$ running through a reduced system of residues modulo $f.$ From (2.3), the number $1+\zeta_{f}^{-k}$ (for $0<k<f$) can be represented as

[TABLE]

which is equivalent to the relation

[TABLE]

Therefore

[TABLE]

Further, since $1+\zeta_{f}^{-k}$ and $1+\zeta_{f}^{k}$ are conjugate, we have

[TABLE]

Now assume that the character $\chi$ (and hence also $\overline{\chi}$) is even. Interchanging $k$ and $-k$ in (2.12), we have

[TABLE]

Combining (2.12), (2.15) (2.16) and (2.17), this yields

[TABLE]

Thus

[TABLE]

since $\sum_{k=1}^{f}\overline{\chi}(k)=0.$ The summation item in (2.19) is unchanged if we replace $k$ by $f-k$. Therefore,

[TABLE]

Then combing with (2.11) and (2.12), we have

[TABLE]

which is the desired result if $\chi$ is a even character.

If the character $\chi$ is odd, then interchanging $k$ and $-k$ in (2.12), we have

[TABLE]

and by (2.15) and (2.16), we have

[TABLE]

and

[TABLE]

The last equality is got by changing $k$ to $-k$ in the second summation and applying the oddness of $\chi$. Then combing with (2.11) and (2.12), we have

[TABLE]

which is the desired result if $\chi$ is an odd character.

3. Proof of Theorem 1.6

For $K^{+}=\mathbb{Q}(\mu_{m})^{+},$ we have $[K^{+}:\mathbb{Q}]=\frac{\varphi(m)}{2}$, where $\varphi$ is Euler-phi function. Let $d$ be the absolute value of the discriminant of $K^{+}$ and $G=\mathbb{Z}(m)^{*}/\{\pm 1\}$ is the Galois group of $K^{+}$ over $\mathbb{Q}$. By the Conductor-Discriminant-formula ([6, p.28 Theorem 3.1]),

[TABLE]

where $\chi$ runs over all the characters of $G$. From the functional equation of Dirichlet L-function, one can deduce that([6, p.36 Corollary 4.6])

[TABLE]

For Re$(s)>1$,

[TABLE]

By analytic continuation, the above equality holds in the whole complex plane. Thus

[TABLE]

Multiplying the class number formula (1.7) by $\eta=\prod_{\chi\neq 1}(1-\chi(2))$ and using (3.4) we have

[TABLE]

where the product over $\chi\neq 1$ is taken over non-trivial characters of $G$, or equivalently, non-trivial even characters of $\mathbb{Z}(m)^{*}$. Applying theorem 1.5,

[TABLE]

Substituting (3.1) and (3.2) into (3.6), we get

[TABLE]

Since

[TABLE]

and there are exactly $\frac{\varphi(m)}{2}$ even characters and $\frac{\varphi(m)}{2}-1$ non-trivial even characters, we have

[TABLE]

which is the desired result.

4. Proof of Proposition 1.7

For $m$ being a prime power, we know that $E=E^{+}W$, where $W=\langle-1,\zeta_{m}\rangle$ is the group of roots of unity in $K=\mathbb{Q}(\zeta_{m})$ ([6, p.40, Theorem 4.12 and Corollary 4.13]). Since $\tilde{\mathcal{E}}\supset W$, the following natural morphism

[TABLE]

induced by the inclusion $E^{+}\rightarrow E$ is surjective. The kernel is $E^{+}\cap\tilde{\mathcal{E}}$, which by definition is just $\tilde{\mathcal{E}}^{+}$. Therefore,

[TABLE]

which is the desired result.

5. Proof of Theorem 1.8

The Galois group of $\mathbb{Q}(\mu_{m})$ over $\mathbb{Q}$ is isomorphic to $\mathbb{Z}(m)^{*}$ under the map:

[TABLE]

where   $\sigma_{a}:\zeta_{m}\mapsto\zeta_{m}^{a}$. If $\epsilon_{1},\ldots,\epsilon_{r}$ is a basis for $E^{+}$ (mod roots of unity), then the regulator $R^{+}$ is the absolute value of the determinant

[TABLE]

where $j=1,\ldots,r$ and $a\in\mathbb{Z}(m)^{*}/\pm 1$ and $a\not\equiv\pm 1$ (mod $m$) (see [4, p. 85]). Let $G=\mathbb{Z}(m)^{*}/\pm 1$ and view $a\in G$ and $a\neq 1$ in $G$. Using the cyclotomic units introduced in (1.13), we also form a new type of cyclotomic regulator as follows

[TABLE]

with $a,k\in G$.

As pointed out by Serre to Lang (see [4, p. 90]), the following determinant relation is due to Dedekind, February 1896, who communicated it to Frobenius in March.

Theorem 5.1** (Dedekind determinant formula, [4, p.90, Theorem 6.2]).**

Let $G$ be a finite group and $f$ be any (complex valued) function on $G$. Then

[TABLE]

Therefore, for a finite abelian group $G$,

[TABLE]

The above Dedekind determinant relation implies the following lemma.

Lemma 5.2**.**

We have for $G=\mathbb{Z}(m)^{*}/\pm 1$,

[TABLE]

Lemma 5.3**.**

Let $m=p^{n}$ be a prime power and $\chi\neq 1$ be a nontrivial character of $G=\mathbb{Z}(m)^{*}/\pm 1$. Denote $G_{\chi}=\mathbb{Z}(f_{\chi})^{*}/\pm 1$. Then, we have

[TABLE]

Proof.

Let $f_{\chi}=p^{s}$. We only need to prove the case $s<n$. Write the residue classes in $\mathbb{Z}(p^{n})$ in the form

[TABLE]

and $k$ ranges over a fixed set of representatives for residue classes of $\mathbb{Z}(p^{s})^{*}$, then we have

[TABLE]

Since

[TABLE]

we have

[TABLE]

which implies that

[TABLE]

For odd $p$, by (5.4) and (5.5), we obtain

[TABLE]

For $p=2$, we have $s\geq 2$ since $\chi\neq 1$. In this case, similarly, we get

[TABLE]

Since $(1+p^{s}/2)^{2}\equiv 1\pmod{p^{s}}$, $\chi(1+p^{s}/2)=\pm 1$. But it can not equal to $1$. Otherwise, the conductor of $\chi$ would be $p^{s}/2$. This explains the last equality.

Summing up, we have

[TABLE]

The minus sign occurs if and only if $p=2$ and $s<n$. Recall

[TABLE]

As $\chi$ is even, dividing both sides of (5.7) by 2, we get the desired formula. ∎

We have the following result related to $h^{+}$ and $\widetilde{R_{\textrm{cyc}}}/R^{+}$.

Proposition 5.4**.**

Let $K=\mathbb{Q}(\mu_{m})$, $K^{+}=\mathbb{Q}(\mu_{m})^{+}$ and $h^{+}$ be the class number of $K^{+}$. Assume $m=p^{n}$ is a prime power. Then

[TABLE]

Proof.

By (5.1), Lemmas 5.2 and 5.3 and Theorem 1.6, we have

[TABLE]

∎

The subgroup $\tilde{\mathcal{E}}^{+}$ has finite index in the group $E^{+}$ if and only if $\widetilde{R_{\textrm{cyc}}}\neq 0$. Moreover, if $\widetilde{R_{\textrm{cyc}}}\neq 0$, by [6, p.41, Lemma 4.15] we have

[TABLE]

From Proposition 5.4, $\widetilde{R_{\textrm{cyc}}}\neq 0$ if and only if $\eta=\prod_{\chi\neq 1}(1-\chi(2))\neq 0$. Clearly, for $m=2^{n}$, $\eta=1$. Therefore, we also have a new formula related to $h^{+}$ and the unit indexes $(E:\tilde{\mathcal{E}})$ and $(E^{+}:\tilde{\mathcal{E}}^{+})$.

Corollary 5.5**.**

Let $K=\mathbb{Q}(\mu_{m})$, $K^{+}=\mathbb{Q}(\mu_{m})^{+}$ and $h^{+}$ be the class number of $K^{+}$. Assume either $m=2^{n}$; or $m=p^{n}$ is an odd prime power and $\chi(2)\neq 1$ for any non-trivial character $\chi$ of $G=\mathbb{Z}(m)^{*}/\pm 1$. Then

[TABLE]

The following lemma concerns the non-vanishing of $\eta$ for the odd case.

Lemma 5.6**.**

For $m=p^{n}$ being an odd prime power, $\chi(2)\neq 1$ for any non-trivial character $\chi$ of $G=\mathbb{Z}(m)^{*}/\pm 1$ if and only if $-1,2$ generate the group $\mathbb{Z}(m)^{*}$.

Proof.

$\chi(2)\neq 1$ for any non-trivial character $\chi$ of $G$ if and only if $\widehat{G/\langle\bar{2}\rangle}$, the character group of $G/\langle\bar{2}\rangle$ is trivial, where $\bar{2}$ is the image of $2$ in $G$, which is equivalent to $G/\langle\bar{2}\rangle$ is trivial. Since $G/\langle\bar{2}\rangle\cong\mathbb{Z}(m)^{*}/\langle-1,2\rangle$, we get the desired result. ∎

Finally by Corollary 5.5 and Lemma 5.6, we get Theorem 1.8.