PROOF OF A CONJECTURE RELATED TO THE PARABOLIC CLASS NUMBERS OF SOME FUCHSIAN GROUPS

This paper proves a conjecture given in [6], which is concerning with the parabolic class numbers of some Fuchsian groups.

The Problem on Class Numbers of Quadratic Number Fields

It is a survey of the problem on class numbers of quadratic number fields.

Some Results Connected with the Class Number Problem in Real Quadratic Fields

We investigate arithmetic properties of certain subsets of square-free positive integers and obtain in this way some results concerning the class number h（d） of the real quadratic field Q（√d）. In particular, we give a new proof of the result of Hasse, asserting that in this case h（d） = 1 is possible only if d is of the form p, 2q or qr. where p.q. r are primes and q≡r≡3（mod 4）.

Quadratic Number Fields with Class Numbers Divisible by a Prime q

Let q 5 be a prime number. Let k d=() be a quadratic number field, where d =--gqq(1)2(1) ---qqqquwuq12((1)+). Then the class number of k is divisible by q for certain integers u,w. Conversely, assume W / k is an unramified cyclic extension of degree q (which implies the class number of k is divisible by q), and W is the splitting field of some irreducible trinomial f(X) = XqaXb with integer coefficients, k Df=(())with D(f) the discriminant of f(X). Then f(X) must be of the form f(X) = Xquq2wXuq1 in a cer-tain sense where u,w are certain integers. Therefore, k d=() with d =-----qqqqqquwuq(1)122(1)((1)+). Moreover, the above two results are both generalized for certain kinds of general polynomials.

Fundamental Unit System and Class Number for Real Number Fields of Type (2,2,2)

Let k=Q((D 2+md)(D 2+nd)(D 2+rd)), this paper proves firstly that the fundamental unit of k is ε=((D 2+md)(D 2+nd)+D 2(D 2+rd)) 2/(|mn|d 2), where D,d,m,n, and r are rational integers satisfying certain conditions. Consequently, we describe the fundamental unit system of K=Q(D 2+md,D 2+nd,D 2+rd) explicitly by the fundamental unit of all the quadratic subfields and the class number h K explicitly by the class numbers of all the quadratic subfields. We also provide the fundamental unit system of some fields of (2,2) type.

Lower Bound for Ideal Class Numbers of Real Quadratic Function Fields

In this paper, the theory of continued fractions of algebraic functions will be used to give a general theorem on lower bounds for class numbers of real quadratic function fields K=k(D). The bounds are given more explicitly for six types of real quadratic function fields. As a consequence, six classes of real quadratic function fields with ideal class number greater than one are given.[

Bounds of the Ideal Class Numbers of Real Quadratic Function Fields

The theory of continued fractions of functions is used to give a lower bound for class numbers h(D) of general real quadratic function fields over k = F q (T). For five series of real quadratic function fields K, the bounds of h(D) are given more explicitly, e. g., if D = F 2 + c, then h(D) ≥ degF/degP; if D = (SG)2 + cS, then h(D) ≥ degS/degP; if D = (A m + a)2 + A, then h(D) ≥ degA/degP, where P is an irreducible polynomial splitting in K, c ∈ F q . In addition, three types of quadratic function fields K are found to have ideal class numbers bigger than one.

A Note on 3-Divisibility of Class Number of Quadratic Field

In this paper,the authors show that there exists infinitely many family of pairs of quadratic fields Q(√D)and Q((√D+n)(1/2))with D,n∈Z whose class numbers are both divisible by 3.

Parametrization of the Quadratic Function Fields Whose Divisor Class Numbers are Divisible by Three

A parametrization of quadratic function fields whose divisor class numbers are divisible by 3 is obtained by using free parameters when the characteristics of the fields are not 3.

TEN FORMULAE OF TYPE ANKENY-ARTINCHOWLA FOR CLASS NUMBERS OF GENERAL CYCLIC QUARTIC FIELDS

Let K be a cyclic quartic number field, and k its quadratic subfield. Let h（L） denote theideal class number of field L. Ten congruenees for h- = h（K）/h（k） are obtained. In par-ticular, if K = Q((p+s(p1/2)))1/2 with the prime number p = r2+s2 and s is even, then C1h-≡B（p-1）/4B3（p-1）/4 （mod p） for p≡1 （mod 8）; and C2h-≡E（p-5）/8E（3p-7）/8（mod p） for p≡5 （mod 8）where Bn and En are the Bernoulli and the Euler numbers. If the real K = Q((v(5+2(51/2)))1/2,then C3h-≡h(Q(（-v）1/2)) h (Q(（-5v）1/2)) （mod 5）. If 3 ramifies in K = Q(θ1/2), then C4h（K）≡h（K*） （mod 3） with K* = Q((-3θ1/2)). All the above Ci are explicitly given constants.Some relations between the factors of class numbers h- are also obtained. These results forcyclic quartic fields are an extension of the results for quadratic fields obtained by Ankeny-Artin-Chowla, Kiselev, Carlitz and Lu Hong-wen from 1948 to 1983.

Congruences for the class numbers of real cyclic sextic number fields

Let K6 be a real cyclic sextic number field, and K2, K3 its quadratic and cubic subfield. Let h（L） denote the ideal class number of field L. Seven congruences for h- = h（K6）/（h（K2）h（K3）） are obtained. In particular, when the conductor f6 of K6 is a prime p, （mod p）, where C is an explicitly given constant, and Bn is the Bernoulli number. These results on real cyclic sextic fields are an extension of the results on quadratic and cyclic quartic fields.

HIRZEBRUCH SUM AND CLASS NUMBER OF THE QUADRATIC FIELDS

1. The purpose of this note is to give the proof of some results in my preprint, For a real quadratic irrational number β,

Criteria of Class Number h(K)=1 for Real Quadratic Number Fields

For real quadratic fields K, especially for fields K of ERD-type, a series of criteria of ideal class numbers h(K)=1 and h(K)】1 will be given via results of Diophantine equations in [1] and continued fraction theory. The problem of class numbers of real quadratic fields, after Gauss’conjecture, has been studied. For example, Lu Hong-wen

Proof of class number formulae by machine --A note on a Chowla's conjecture

As an attempt to follow the direction of machine proof, a personal computer LEO386/25 is used to prove some class number formulae for certain imaginary quadratic number fields Q (p) , ( q =3, 7, 11, 19, 23, 31, 43 and 47) if the real quadratic number field Q (-p) has class number one for a prime p=4N 2+1 (N is a positive integer).

Class numbers of cyclic 2-extensions and Gross conjecture over Q

The Gross conjecture over Q was first claimed by Aoki in 1991.However,the original proof contains too many mistakes and false claims to be considered as a serious proof.This paper is an attempt to find a sound proof of the Gross conjecture under the outline of Aoki.We reduce the conjecture to an elementary conjecture concerning the class numbers of cyclic 2-extensions of Q.

Class number relation between type (l,l/,…, l) function fields over F_q(T) and their subfields

Let L/Fq(T) be a tame abelian extension of type (l, l,...l). The ratio of the degree zero divisor dass number (as well as the ideal class number) of L to the product of corresponding class numbers of all cydic subfields of L is clearly determined.

CONGRUENCES FOR CLASS NUMBERS OF GENERAL CYCLIC QUARTIC FIELDS

Ankeny-Artin-Chowla obtained in [1] several congruences for class number h of quadratic number field k, some of which were obtained also by Kiselev. In particular, if the discriminant of k is a prime p≡1（mod 4） and ε0=(t+p1/u)/2 is the fundamental unit of k,

A Note on Asymptotic Class Number Upper Bounds in p-adic Lie Extensions

Let p be an odd prime and F∞ a p-adic Lie extension of a number field F with Galois group G. Suppose that G is a compact pro-p p-adic Lie group with no torsion and that it contains a closed normal subgroup H such that G/H≌Zp. Under various assumptions, we establish asymptotic upper bounds for the growth of p-exponents of the class groups in the said p-adic Lie extension. Our results generalize a previous result of Lei, where he established such an estimate under the assumption that H≌Zp.

Congruence formulae modulo powers of 2 for class numbers of cyclic quartic fields

Let K = k(√θ) be a real cyclic quartic field, k be its quadratic subfield and K = k(√θ) be the corresponding imaginary quartic field. Denote the class numbers of K, k and K by hK, hk and hK respectively. Here congruences modulo powers of 2 for h = hK /hk and h = hK /hk are obtained via studying the p-adic L-functions of the fields.

STABILITY NUMBER IN SUBCLASSES OF P_5^-FREE GRAPHS

Two new hereditary classes of P 5-free graphs where the stability number can be found in polynomial time are proposed.They generalize several known results.