Explicit bounds for class numbers ================================= .. if-builder:: html .. toctree:: :maxdepth: 2 Let :math:`K` be a number field of degree :math:`n\ge2`, signature :math:`(r_1,r_2)`, absolute value of discriminant :math:`d_K`, class number :math:`h_K`, regulator :math:`\mathcal{R}_K` and :math:`w_K` the number of roots of unity in :math:`K`. We further denote by :math:`\kappa_K` the residue at :math:`s=1` of the Dedekind zeta-function :math:`\zeta_K(s)` attached to :math:`K`. Estimating :math:`h_K` is a long-standing problem in algebraic number theory. 1. Majorising :math:`h_K\mathcal{R}_K` -------------------------------------- One of the classic way is the use of the so-called analytic class number formula stating that :math:`\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 :math:`\kappa_K`. This is done in :cite:`Louboutin00` and in :cite:`Louboutin01` with additional properties of log-convexity of some functions related to :math:`\zeta_K` and enabled Louboutin to reach the following bound: :math:`\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 :math:`\zeta_K(\beta)=0` for some :math:`\tfrac12\le \beta< 1`, then we have :math:`\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 :math:`K` is abelian, then the residue :math:`\kappa_K` may be expressed as a product of values at :math:`s=1` of :math:`L`-functions associated to primitive Dirichlet characters attached to :math:`K`. On using estimates for such :math:`L`-functions from :cite:`Ramare01`, we get for instance :math:`\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 :math:`\frac14(5-\log 36)=0.354\cdots` can be improved upon in many cases. For instance, when :math:`K` is abelian and totally real (i.e. :math:`r_2=0`), a result fromd :cite:`Ramare01` implies that the constant may be replaced by 0, so that :math:`\displaystyle h_K\mathcal{R}_K\le\left(\frac{\log d_K}{4n-4}\right)^{n-1}\sqrt{d_K}.` 2. Majorising :math:`h_K` ------------------------- One may also estimate :math:`h_K` alone, without any contamination by the regulator since this contamination is often difficult to control, see :cite:`Pohst-Zassenhaus89`. In this case, one rather uses explicit bounds for the Piltz-Dirichlet divisor functions :math:`\tau_n` (see :cite:`Bordelles02` and :cite:`Bordelles06`) and get :math:`\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 :math:`\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 :math:`M_K` is known as the Minkowski constant of K. In :cite:`Cully-Trudgian21` we find the following. .. admonition:: Theorem (2021) :class: thm-tme-emt Ler :math:`K` be a quartic number field with class number :math:`h_K` and Minkowski bound :math:`b`. Then if :math:`b\ge 193`, we have :math:`h_K\le (1/3) x(\log x)^3`. 3. Using the influence of the small primes ------------------------------------------ It is explained in :cite:`Louboutin05` how the behavior of certain small primes may subtantially improve on the previous bounds. To make things more significant, define, for a rational prime :math:`p`, :math:`\displaystyle \Pi_K(p)=\prod_{\mathfrak{p}|p}\left(1-\frac{1}{\mathcal{N}_K(\mathfrak{p})}\right)^{-1}.` From :cite:`Louboutin05`, we have among other things :math:`\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 :math:`K` is any number field of degree :math:`n\ge3`. In particular, when :math:`2` is inert in :math:`K`, then :math:`\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 :math:`h^-_K` of CM-fields ---------------------------------- Let :math:`K` be here a CM-field of degree :math:`2n > 2`, i.e. a totally complex quadratic extension :math:`K` of its maximal totally real subfield :math:`K^+`. it is well known that :math:`h_{K^+}` divides :math:`h_K`. The quotient is denoted by :math:`h^-_K` and is called the relative class number of :math:`K`. The analytic class number formula yields :math:`\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 :math:`\chi` is the quadratic character of degree 1 attached to the extension :math:`K/K^+` and :math:`Q_K\in\{1,2\}` is the Hasse unit index of :math:`K`. Here are three results originating in this formula. From :cite:`Louboutin00`: .. admonition:: Theorem (2000) :class: thm-tme-emt We have :math:`\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 :cite:`Louboutin03`: .. admonition:: Theorem (2003) :class: thm-tme-emt Assume that :math:`(\zeta_K/\zeta_{K^+})(\sigma)\ge0` whenever :math:`0 < \sigma < 1`. Then we have :math:`\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 :cite:`Louboutin03`: .. admonition:: Theorem (2003) :class: thm-tme-emt Let :math:`c=6-4\sqrt{2}=0.3431\cdots`. Assume that :math:`d_K\ge 2800^n` and that either :math:`K` does not contain any imaginary quadratic subfield, or that the real zeros in the range :math:`1-\frac{c}{\log d_N}\le \sigma < 1` of the Dedekind zeta-functions of the imaginary quadratic subfields of :math:`K` are nor zeros of :math:`\zeta_K(s)`, where :math:`N` is the normal closure of :math:`K`. Then we have :math:`\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 :cite:`Louboutin03`: .. admonition:: Theorem (2003) :class: thm-tme-emt Assume :math:`n > 2`, :math:`d_K > 2800^n` and that :math:`K` contains an imaginary quadratic subfield :math:`F` such that :math:`\zeta_F(\beta)=\zeta_K(\beta)=0` for some :math:`\beta` satisfying :math:`1-\frac{2}{\log d_K}\le \beta < 1`. Then we have :math:`\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}.`