Explicit bounds for class numbers

Let K be a number field of degree n2, signature (r1,r2), absolute value of discriminant dK, class number hK, regulator RK and wK the number of roots of unity in K. We further denote by κK the residue at s=1 of the Dedekind zeta-function ζK(s) attached to K.

Estimating hK is a long-standing problem in algebraic number theory.
1. Majorising hKRK
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. This is done in [Louboutin, 2000 †Louboutin, S. 2000
Explicit bounds for residues of Dedekind zeta functions, values of {L}-functions at {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}-functions at {s=1}, and explicit lower bounds for relative class numbers of CM-fields
Canad. J. Math., 53(6), 1194--1222.
] with additional properties of log-convexity of some functions related to ζK and enabled Louboutin to reach the following bound:
hKRKwK2(2π)r2(elogdK4n4)n1dK.
Furthermore, if ζK(β)=0 for some 12β<1, then we have
hKRK(1β)wK(2π)r2(elogdK4n)ndK.
When K is abelian, then the residue κK 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 †Ramaré, O. 2001
Approximate Formulae for L(1,χ)
Acta Arith., 100, 245--266.
], we get for instance
hKRKwK2(2π)r2(logdK4n4+5log364)n1dK.
Note that the constant 14(5log36)=0.354 can be improved upon in many cases. For instance, when K is abelian and totally real (i.e. r2=0), a result from [Ramaré, 2001 †Ramaré, O. 2001
Approximate Formulae for L(1,χ)
Acta Arith., 100, 245--266.
] implies that the constant may be replaced by 0, so that
hKRK(logdK4n4)n1dK.
2. Majorising hK
One may also estimate hK 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 (see [Bordellès, 2002 †Bordellès, O. 2002
Explicit upper bounds for the average order of {dn(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 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,
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 is any number field of degree n3. In particular, when 2 is inert in K, then
hKRKwK2(2n1)(2π)r2(elogdK4n4×enlog4/logdK)n1dK.
4. The hK 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 hK+ divides hK. The quotient is denoted by hK and is called the relative class number of K. The analytic class number formula yields
hK=QKwK(2π)n(dKdK+)1/2κKκK+=QKwK(2π)n(dKdK+)1/2L(1,χ)
where χ is the quadratic character of degree 1 attached to the extension K/K+ and QK{1,2} is the Hasse unit index of 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}-functions at {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} 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 whenever 0<σ<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} 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. Assume that dK2800n and that either K does not contain any imaginary quadratic subfield, or that the real zeros in the range 1clogdNσ<1 of the Dedekind zeta-functions of the imaginary quadratic subfields of K are nor zeros of ζK(s), where N is the normal closure of 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} 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, dK>2800n and that K contains an imaginary quadratic subfield F such that ζF(β)=ζK(β)=0 for some β satisfying 12logdKβ<1. Then we have
hK6(πe)2(dKdK+)1/21/n(nπelogdK)n1.

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