Explicit bounds for class numbers

Let K$K$ be a number field of degree n2$n\ge2$, signature (r1,r2)$(r_1,r_2)$, absolute value of discriminant dK$d_K$, class number hK$h_K$, regulator RK$\mathcal{R}_K$ and wK$w_K$ the number of roots of unity in K$K$. We further denote by κK$\kappa_K$ the residue at s=1$s=1$ of the Dedekind zeta-function ζK(s)$\zeta_K(s)$ attached to K$K$.

Estimating hK$h_K$ is a long-standing problem in algebraic number theory.
1. Majorising hKRK$h_K\mathcal{R}_K$
One of the classic way is the use of the so-called analytic class number formula stating that
hKRK=wKdK2r1(2π)r2κK
and to use Hecke's integral representation of the Dedekind zeta function to bound κK$\kappa_K$. This is done in [Louboutin, 2000 †Louboutin, S. 2000
Explicit bounds for residues of Dedekind zeta functions, values of {L$L$}-functions at {s=1$s=1$}, and relative class numbers
J. Number Theory, 85(2), 263--282.
] and in [Louboutin, 2001 †Louboutin, S. 2001
Explicit upper bounds for residues of Dedekind zeta functions and values of {L$L$}-functions at {s=1$s=1$}, and explicit lower bounds for relative class numbers of CM-fields
] with additional properties of log-convexity of some functions related to ζK$\zeta_K$ and enabled Louboutin to reach the following bound:
hKRKwK2(2π)r2(elogdK4n4)n1dK.
Furthermore, if ζK(β)=0$\zeta_K(\beta)=0$ for some 12β<1$\tfrac12\le \beta< 1$, then we have
hKRK(1β)wK(2π)r2(elogdK4n)ndK.
When K$K$ is abelian, then the residue κK$\kappa_K$ may be expressed as a product of values at s=1$s=1$ of L$L$-functions associated to primitive Dirichlet characters attached to K$K$. On using estimates for such L$L$-functions from [Ramaré, 2001 †Ramaré, O. 2001
Approximate Formulae for L(1,χ)$L(1,\chi)$
Acta Arith., 100, 245--266.
], we get for instance
hKRKwK2(2π)r2(logdK4n4+5log364)n1dK.
Note that the constant 14(5log36)=0.354$\frac14(5-\log 36)=0.354\cdots$ can be improved upon in many cases. For instance, when K$K$ is abelian and totally real (i.e. r2=0$r_2=0$), a result from [Ramaré, 2001 †Ramaré, O. 2001
Approximate Formulae for L(1,χ)$L(1,\chi)$
Acta Arith., 100, 245--266.
] implies that the constant may be replaced by 0, so that
hKRK(logdK4n4)n1dK.
2. Majorising hK$h_K$
One may also estimate hK$h_K$ alone, without any contamination by the regulator since this contamination is often difficult to control, see [Pohst & Zassenhaus, 1989 †Pohst, M., & Zassenhaus, H. 1989
Algorithmic algebraic number theory
Encyclopedia of Mathematics and its Applications, vol. 30. Cambridge: Cambridge University Press.
]. In this case, one rather uses explicit bounds for the Piltz-Dirichlet divisor functions τn$\tau_n$ (see [Bordellès, 2002 †Bordellès, O. 2002
Explicit upper bounds for the average order of {dn(m)$d_n(m)$} and application to class number
JIPAM. J. Inequal. Pure Appl. Math., 3(3), Article 38, 15 pp. (electronic).
] and [Bordellès, 2006 †Bordellès, O. 2006
An inequality for the class number
JIPAM. J. Inequal. Pure Appl. Math., 7(3), Article 87, 8 pp. (electronic).
]) and get
hKMK(n1)!(log(M2KdK)2+n2)n1dK
as soon as
n3,dK139M2KwhereMK=(4/π)r2n!/nn.
The constant MK$M_K$ is known as the Minkowski constant of K.
3. Using the influence of small primes
It is explained in [Louboutin, 2005 †Louboutin, S.R. 2005
On the use of explicit bounds on residues of Dedekind zeta functions taking into account the behavior of small primes
J. Théor. Nombres Bordeaux, 17(2), 559--573.
] 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$p$,
ΠK(p)=p|p(11NK(p))1.
From [Louboutin, 2005 †Louboutin, S.R. 2005
On the use of explicit bounds on residues of Dedekind zeta functions taking into account the behavior of small primes
J. Théor. Nombres Bordeaux, 17(2), 559--573.
], we have among other things
hKRKwK2(2π)r2ΠK(2)ΠQ(2)n(elogdK4n4×enlog4/logdK)n1dK
where K$K$ is any number field of degree n3$n\ge3$. In particular, when 2$2$ is inert in K$K$, then
hKRKwK2(2n1)(2π)r2(elogdK4n4×enlog4/logdK)n1dK.
4. The hK$h^-_K$ of CM-fields
Let K$K$ be here a CM-field of degree 2n>2$2n > 2$, i.e. a totally complex quadratic extension K$K$ of its maximal totally real subfield K+$K^+$. it is well known that hK+$h_{K^+}$ divides hK$h_K$. The quotient is denoted by hK$h^-_K$ and is called the relative class number of K$K$. The analytic class number formula yields
hK=QKwK(2π)n(dKdK+)1/2κKκK+=QKwK(2π)n(dKdK+)1/2L(1,χ)
where χ$\chi$ is the quadratic character of degree 1 attached to the extension K/K+$K/K^+$ and QK{1,2}$Q_K\in\{1,2\}$ is the Hasse unit index of K$K$. Here are three results originating in this formula. From [Louboutin, 2000 †Louboutin, S. 2000
Explicit bounds for residues of Dedekind zeta functions, values of {L$L$}-functions at {s=1$s=1$}, and relative class numbers
J. Number Theory, 85(2), 263--282.
]:
Theorem (2000)
We have
hK2QKwK(dKdK+)1/2(elog(dK/dK+)4πn)n.
From [Louboutin, 2003 †Louboutin, S. 2003
Explicit lower bounds for residues at {s=1$s=1$} of Dedekind zeta functions and relative class numbers of CM-fields
Trans. Amer. Math. Soc., 355(8), 3079--3098 (electronic).
]:
Theorem (2003)
Assume that (ζK/ζK+)(σ)0$(\zeta_K/\zeta_{K^+})(\sigma)\ge0$ whenever 0<σ<1$0 < \sigma < 1$. Then we have
hKQKwKπelogdK(dKdK+)1/2(n1πelogdK)n1.
Again from [Louboutin, 2003 †Louboutin, S. 2003
Explicit lower bounds for residues at {s=1$s=1$} of Dedekind zeta functions and relative class numbers of CM-fields
Trans. Amer. Math. Soc., 355(8), 3079--3098 (electronic).
]:
Theorem (2003)
Let c=642=0.3431$c=6-4\sqrt{2}=0.3431\cdots$. Assume that dK2800n$d_K\ge 2800^n$ and that either K$K$ does not contain any imaginary quadratic subfield, or that the real zeros in the range 1clogdNσ<1$1-\frac{c}{\log d_N}\le \sigma < 1$ of the Dedekind zeta-functions of the imaginary quadratic subfields of K$K$ are nor zeros of ζK(s)$\zeta_K(s)$, where N$N$ is the normal closure of K$K$. Then we have
hKcQKwK4nec/2[N:Q](dKdK+)1/2(nπelogdK)n.
And a third result from [Louboutin, 2003 †Louboutin, S. 2003
Explicit lower bounds for residues at {s=1$s=1$} of Dedekind zeta functions and relative class numbers of CM-fields
Trans. Amer. Math. Soc., 355(8), 3079--3098 (electronic).
]:
Theorem (2003)
Assume n>2$n > 2$, dK>2800n$d_K > 2800^n$ and that K$K$ contains an imaginary quadratic subfield F$F$ such that ζF(β)=ζK(β)=0$\zeta_F(\beta)=\zeta_K(\beta)=0$ for some β$\beta$ satisfying 12logdKβ<1$1-\frac{2}{\log d_K}\le \beta < 1$. Then we have
hK6(πe)2(dKdK+)1/21/n(nπelogdK)n1.

Last updated on August 23rd, 2012, by Olivier Bordellès