Multiple Dedekind sums and relative class number formulae

We construct some multiple Dedekind sums and relate them to the relative class number of an imaginary abelian number field. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Heuristics and related results on class groups of real quadratic fields

The fact is studied that the ideal class numbersh of types of real quadratic fields usually contain a fixed prime numberp as a factor, and the reason is found to be existing there a kind of prime ideals whosepth powers are principal. A modification of the Cohen-Lenstra Heuristics for the probability that in this situation the class numberh is actually a multiple ofp then is presented: Prob (p|h)=1-(1-p ^-1)(1-P ^-2)⋯. This idea is also extended to predict the probability that the classP represented by the above prime ideal is actually of orderp: Prob (_o(P)=p) =1/p. Both of these predictions agree fairly well with the numerical data. Project supported by the National Natural Science Foundation of China.     

On the number of real quadratic fields with class number divisible by 3

We find a lower bound for the number of real quadratic fields whose class groups have an element of order . More precisely, we establish that the number of real quadratic fields whose absolute discriminant is and whose class group has an element of order is improving the existing best known bound of R. Murty.

     

The tame kernel of imaginary quadratic fields with class number 2 or 3

This paper presents improved bounds for the norms of exceptional finite places of the group , where is an imaginary quadratic field of class number 2 or 3. As an application we show that .

     

On class number relations in characteristic two

A theorem of F. Hirzebruch relates continued fractions to class numbers of quadratic number fields. A version for function fields of odd characteristic was established by D. R. Hayes and C. D. González. We present here a complete treatment of the even characteristic theory, in particular, two class number relations involving continued fractions are derived, one of which is an analogue of the Hirzebruch relation in characteristic 2. Research partially supported by National Science Council, Republic of China.     

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.     

On quasi-reduced quadratic forms

With the help of continued fractions, we plan to list all the elements of the set Q△ = {aX2 ＋ bXY ＋ cY2 ： a,b, c ∈Z, b2 - 4ac = △ with 0 ≤ b 〈 √△}of quasi-reduced quadratic forms of fundamental discriminant △. As a matter of fact, we show that for each reduced quadratic form f = aX2 ＋ bXY ＋ cY2 = （a, b, c） of discriminant △〉0（and of sign σ（f） equal to the sign of a）, the quadratic forms associated with f and defined by {〈a＋bu＋cu2,b＋2cu.c〉},with 1≤σ（f）u≤b/2｜c｜ （whenever they exist）, 〈c,-b-2cu,a＋bu＋cu2〉 with b/2｜c｜≤σ（f）u≤[w（f）]=[b＋√△/2｜c｜], are all different from one another and build a set I（f） whose cardinality is #I（f）={1＋[ω（f）],when（2c）｜b,[ω（f）],when （2c）｜b. If f and g are two different reduced quadratic forms, we show that I（f） ∩ I（g） = Ф. Our main result is that the set Q△ is given by the disjoint union of all I（f） with f running through the set of reduced quadratic forms of discriminant △〉0. This allows us to deduce a formula for #（Q△） involving sums of partial quotients of certain continued fractions.     

The Problem on Class Numbers of Quadratic Number Fields

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

Computation of class numbers of quadratic number fields

We explain how one can dispense with the numerical computation of approximations to the transcendental integral functions involved when computing class numbers of quadratic number fields. We therefore end up with a simpler and faster method for computing class numbers of quadratic number fields. We also explain how to end up with a simpler and faster method for computing relative class numbers of imaginary abelian number fields.

     

Subgroups of ideal class groups of real quadratic algebraic function fields

Necessary and sufficient condition on real quadratic algebraic function fields K is given for their ideal class groups H(K) to contain cyclic subgroups of order n. And eight series of such real quadratic function fields K are obtained whose ideal class groups contain cyclic subgroups of order n. In particular, the ideal class numbers of these function fields are divisible by n.     

Computing the rank of elliptic curves over real quadratic number fields of class number 1

In this paper we describe an algorithm for computing the rank of an elliptic curve defined over a real quadratic field of class number one. This algorithm extends the one originally described by Birch and Swinnerton-Dyer for curves over . Several examples are included.

     

Classification of integral lattices with large class number

A detailed exposition of Kneser's neighbour method for quadratic lattices over totally real number fields, and of the sub-procedures needed for its implementation, is given. Using an actual computer program which automatically generates representatives for all isomorphism classes in one genus of rational lattices, various results about genera of -elementary lattices, for small prime level are obtained. For instance, the class number of -dimensional -elementary even lattices of determinant is ; no extremal lattice in the sense of Quebbemann exists. The implementation incorporates as essential parts previous programs of W. Plesken and B. Souvignier.

     

Class number bounds and Catalan's equation

We improve a criterion of Inkeri and show that if there is a solution to Catalan's equation

with and prime numbers greater than 3 and both congruent to 3 , then and form a double Wieferich pair. Further, we refine a result of Schwarz to obtain similar criteria when only one of the exponents is congruent to 3 . Indeed, in light of the results proved here it is reasonable to suppose that if , then and form a double Wieferich pair.

     

实二次域的一个同余式

本文证明了关于实二次域的类数和某类特征和的同余式,同时给出某类实二次域的类数可除性的一个判别法则．     

Class number 5, 6 and 7

We outline the determination of all imaginary quadratic fields with class number 5, 6 or 7.

     

Imaginary quadratic fields with noncyclic ideal class groups

Let g be an odd positive integer and X be a positive real number. We shall show that for any ∊ > 0, the number of imaginary quadratic fields with discriminant ≥ − X and ideal class group having a subgroup isomorphic to ℤ/g ℤ× ℤ/g ℤ is ≫ X^1/g−∊. 2000 Mathematics Subject Classification Primary—11R11, 11R29 This work was supported by KRF-R08-2003-000-10243-0 and partially by KRF-2002-003-C00001.     

Approximations of e and π: an exploration

ABSTRACT

Fractional approximations of e and π are discovered by searching for repetitions or partial repetitions of digit strings in their expansions in different number bases. The discovery of such fractional approximations is suggested for students and teachers as an entry point into mathematics research.     

Ideal class groups and their subgroups of real quadratic fields

A necessary and sufficient condition is given for the ideal class group H(m) of a real quadratic field Q (√m) to contain a cyclic subgroup of ordern. Some criteria satisfying the condition are also obtained. And eight types of such fields are proved to have this property, e.g. fields withm=(z ⁿ +t−1)²+4t(witht|z ⁿ −1), which contains the well-known fields withm=4z ⁿ +1 andm=4z ²ⁿ +4 as special cases. Project supported by the National Natural Science Foundation of China.     

Units in some families of algebraic number fields

Multi-dimensional continued fractions associated with are introduced and applied to find systems of fundamental units in some families of totally real fields and fields with signature (2,1).

     

Congruences Involving Bernoulli Numbers and Fermat-Euler Quotients

Let B_m be the mth Bernoulli number in the even suffix notation and let q(a, n)=(a^?(n)−1)/n be the Fermat-Euler quotient, where a, n?2 are relatively prime positive integers and ? is the Euler totient function. The main purpose of this paper is to devise a certain congruence involving the Bernoulli number and Fermat-Euler quotient, which leads to several important arithmetic properties of Bernoulli numbers.