Explicit bounds for class numbers

Let \(K\) be a number field of degree \(n\ge2\), signature \((r_1,r_2)\), absolute value of discriminant \(d_K\), class number \(h_K\), regulator \(\mathcal{R}_K\) and \(w_K\) the number of roots of unity in \(K\). We further denote by \(\kappa_K\) the residue at \(s=1\) of the Dedekind zeta-function \(\zeta_K(s)\) attached to \(K\).

Estimating \(h_K\) is a long-standing problem in algebraic number theory.

1. Majorising \(h_K\mathcal{R}_K\)

One of the classic way is the use of the so-called analytic class number formula stating that

\(\displaystyle h_K\mathcal{R}_K=\frac{w_K \sqrt{d_K}}{2^{r_1}(2\pi)^{r_2}}\kappa_K\) and to use Hecke’s integral representation of the Dedekind zeta function to bound \(\kappa_K\). This is done in [Louboutin, 2000] and in [Louboutin, 2001] with additional properties of log-convexity of some functions related to \(\zeta_K\) and enabled Louboutin to reach the following bound:

\(\displaystyle h_K\mathcal{R}_K\le\frac{w_K}{2}\left(\frac{2}{\pi}\right)^{r_2}\left(\frac{e\log d_K}{4n-4}\right)^{n-1}\sqrt{d_K}.\) Furthermore, if \(\zeta_K(\beta)=0\) for some \(\tfrac12\le \beta< 1\), then we have

\(\displaystyle h_K\mathcal{R}_K\le(1-\beta)w_K\left(\frac{2}{\pi}\right)^{r_2}\left(\frac{e\log d_K}{4n}\right)^{n}\sqrt{d_K}.\) When \(K\) is abelian, then the residue \(\kappa_K\) may be expressed as a product of values at \(s=1\) of \(L\)-functions associated to primitive Dirichlet characters attached to \(K\). On using estimates for such \(L\)-functions from [Ramaré, 2001], we get for instance

\(\displaystyle h_K\mathcal{R}_K\le\frac{w_K}{2}\left(\frac{2}{\pi}\right)^{r_2}\left(\frac{\log d_K}{4n-4}+\frac{5-\log 36}{4}\right)^{n-1}\sqrt{d_K}.\) Note that the constant \(\frac14(5-\log 36)=0.354\cdots\) can be improved upon in many cases. For instance, when \(K\) is abelian and totally real (i.e. \(r_2=0\)), a result fromd [Ramaré, 2001] implies that the constant may be replaced by 0, so that

\(\displaystyle h_K\mathcal{R}_K\le\left(\frac{\log d_K}{4n-4}\right)^{n-1}\sqrt{d_K}.\)

2. Majorising \(h_K\)

One may also estimate \(h_K\) alone, without any contamination by the regulator since this contamination is often difficult to control, see [Pohst and Zassenhaus, 1989].

In this case, one rather uses explicit bounds for the Piltz-Dirichlet divisor functions \(\tau_n\) (see [Bordellès, 2002] and [Bordellès, 2006]) and get

\(\displaystyle h_K\le \frac{M_K}{(n-1)!}\left(\frac{\log\bigl(M^2_Kd_K\bigr)}{2}+n-2\right)^{n-1}\sqrt{d_K}\) as soon as

\(\displaystyle n\ge 3,\quad d_K\ge 139 M_K^{-2}\quad\text{where}\quad M_K=(4/\pi)^{r_2}n!/n^n.\) The constant \(M_K\) is known as the Minkowski constant of K.

In [Cully-Hugill and Trudgian, 2021] we find the following.

Theorem (2021)

Ler \(K\) be a quartic number field with class number \(h_K\) and Minkowski bound \(b\). Then if \(b\ge 193\), we have \(h_K\le (1/3) x(\log x)^3\).

3. Using the influence of the small primes

It is explained in [Louboutin, 2005] how the behavior of certain small primes may subtantially improve on the previous bounds. To make things more significant, define, for a rational prime \(p\),

\(\displaystyle \Pi_K(p)=\prod_{\mathfrak{p}|p}\left(1-\frac{1}{\mathcal{N}_K(\mathfrak{p})}\right)^{-1}.\) From [Louboutin, 2005], we have among other things

\(\displaystyle h_K\mathcal{R}_K\le\frac{w_K}{2}\left(\frac{2}{\pi}\right)^{r_2}\frac{\Pi_K(2)}{\Pi_{\mathbb{Q}}(2)^n}\left(\frac{e\log d_K}{4n-4}\times e^{n\log 4/\log d_K}\right)^{n-1}\sqrt{d_K}\) where \(K\) is any number field of degree \(n\ge3\). In particular, when \(2\) is inert in \(K\), then

\(\displaystyle h_K\mathcal{R}_K\le\frac{w_K}{2(2^n-1)}\left(\frac{2}{\pi}\right)^{r_2}\left(\frac{e\log d_K}{4n-4}\times e^{n\log 4/\log d_K}\right)^{n-1}\sqrt{d_K}.\)

4. The \(h^-_K\) of CM-fields

Let \(K\) be here a CM-field of degree \(2n > 2\), i.e. a totally complex quadratic extension \(K\) of its maximal totally real subfield \(K^+\). it is well known that \(h_{K^+}\) divides \(h_K\). The quotient is denoted by \(h^-_K\) and is called the relative class number of \(K\). The analytic class number formula yields

\(\displaystyle h^-_K=\frac{Q_Kw_K}{(2\pi)^n}\left(\frac{d_K}{d_{K^+}}\right)^{1/2}\frac{\kappa_K}{\kappa_{K^+}}=\frac{Q_Kw_K}{(2\pi)^n}\left(\frac{d_K}{d_{K^+}}\right)^{1/2}L(1,\chi)\) where \(\chi\) is the quadratic character of degree 1 attached to the extension \(K/K^+\) and \(Q_K\in\{1,2\}\) is the Hasse unit index of \(K\). Here are three results originating in this formula.

From [Louboutin, 2000]:

Theorem (2000)

We have \(\displaystyle h^-_K \le 2Q_Kw_K\left(\frac{d_K}{d_{K^+}}\right)^{1/2} \left( \frac{e\log(d_K/d_{K^+})}{4\pi n} \right)^n.\)

From [Louboutin, 2003]:

Theorem (2003)

Assume that \((\zeta_K/\zeta_{K^+})(\sigma)\ge0\) whenever \(0 < \sigma < 1\). Then we have \(\displaystyle h^-_K \ge \frac{Q_Kw_K}{\pi e \log d_K} \left(\frac{d_K}{d_{K^+}}\right)^{1/2} \left( \frac{n-1}{\pi e\log d_K} \right)^{n-1}.\)

Again from [Louboutin, 2003]:

Theorem (2003)

Let \(c=6-4\sqrt{2}=0.3431\cdots\). Assume that \(d_K\ge 2800^n\) and that either \(K\) does not contain any imaginary quadratic subfield, or that the real zeros in the range \(1-\frac{c}{\log d_N}\le \sigma < 1\) of the Dedekind zeta-functions of the imaginary quadratic subfields of \(K\) are nor zeros of \(\zeta_K(s)\), where \(N\) is the normal closure of \(K\). Then we have \(\displaystyle h^-_K \ge \frac{cQ_Kw_K}{4ne^{c/2}[N:\mathbb{Q}]} \left(\frac{d_K}{d_{K^+}}\right)^{1/2} \left( \frac{n}{\pi e\log d_K} \right)^{n}.\)

And a third result from [Louboutin, 2003]:

Theorem (2003)

Assume \(n > 2\), \(d_K > 2800^n\) and that \(K\) contains an imaginary quadratic subfield \(F\) such that \(\zeta_F(\beta)=\zeta_K(\beta)=0\) for some \(\beta\) satisfying \(1-\frac{2}{\log d_K}\le \beta < 1\). Then we have \(\displaystyle h^-_K \ge \frac{6}{(\pi e)^2} \left(\frac{d_K}{d_{K^+}}\right)^{1/2-1/n} \left( \frac{n}{\pi e\log d_K} \right)^{n-1}.\)