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.     

实二次域上理想类的Zeta-函数在-3处的值

用简单连分数给出了实二次域理想类的Zeta-函数在-3处值的一个具体的计算公式.     

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.

     

On the exceptional fields for a class of real quadratic fields

In this paper, we give a lower bound exp(2.2 × 10~8 ) for those discriminants of real quadratic fields Q(√ d) with d= N~2-4 and h(d)=1.     

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.     

Quadratic function fields with exponent two ideal class group

It has been shown by Madden that there are only finitely many quadratic extensions of k(x), k a finite field, in which the ideal class group has exponent two and the infinity place of k(x) ramifies. We give a characterization of such fields that allow us to find a full list of all such field extensions for future reference. In doing so we correct some errors in earlier published literature.     

On a class of elliptic functions associated with imaginary quadratic fields

Let be the field discriminant of an imaginary quadratic field. We construct a class of elliptic functions associated naturally with the quadratic field which, combined with the general theory of elliptic functions, allows us to provide a unified theory for two fundamental results (one classical and one due to Ramanujan) about the elliptic functions.

     

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 .

     

The exponent three class group problem for some real cyclic cubic number fields

We determine all the simplest cubic fields whose ideal class groups have exponent dividing , thus generalizing the determination by G. Lettl of all the simplest cubic fields with class number and the determination by D. Byeon of all all the simplest cubic fields with class number . We prove that there are simplest cubic fields with ideal class groups of exponent (and simplest cubic fields with ideal class groups of exponent , i.e. with class number one).

     

Computations of class numbers of real quadratic fields

In this paper an unconditional probabilistic algorithm to compute the class number of a real quadratic field is presented, which computes the class number in expected time . The algorithm is a random version of Shanks' algorithm. One of the main steps in algorithms to compute the class number is the approximation of . Previous algorithms with the above running time , obtain an approximation for by assuming an appropriate extension of the Riemann Hypothesis. Our algorithm finds an appoximation for without assuming the Riemann Hypothesis, by using a new technique that we call the `Random Summation Technique'. As a result, we are able to compute the regulator deterministically in expected time . However, our estimate of on the running time of our algorithm to compute the class number is not effective.

     

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.     

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.     

Hyperelliptic curves and homomorphisms to ideal class groups of quadratic number fields

For the function field K of hyperelliptic curves over Q we define a subgroup of the ideal class group called the group of Z-primitive ideals. We then show that there are homomorphisms from this subgroup to ideal class groups of certain quadratic number fields.     

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.

     

Computing discrete logarithms in real quadratic congruence function fields of large genus

The discrete logarithm problem in various finite abelian groups is the basis for some well known public key cryptosystems. Recently, real quadratic congruence function fields were used to construct a public key distribution system. The security of this public key system is based on the difficulty of a discrete logarithm problem in these fields. In this paper, we present a probabilistic algorithm with subexponential running time that computes such discrete logarithms in real quadratic congruence function fields of sufficiently large genus. This algorithm is a generalization of similar algorithms for real quadratic number fields.

     

Congruences for the class numbers of real cyclic sextic number fields

LetK ₆ be a real cyclic sextic number field, andK ₂,K ₃ its quadratic and cubic subfield. Leth(L) denote the ideal class number of fieldL. Seven congruences forh ^- =h (K ₆)/(h(K ₂)h(K ₃)) are obtained. In particular, when the conductorf ₆ ofK ₆ is a primep, , whereC is an explicitly given constant, andB _n is the Bernoulli number. These results on real cyclic sextic fields are an extension of the results on quadratic and cyclic quartic fields. Project supported by the National Natural Science Foundation of China (Grant No. 19771052).     

Efficient computation of root numbers and class numbers of parametrized families of real abelian number fields

Let be a parametrized family of simplest real cyclic cubic, quartic, quintic or sextic number fields of known regulators, e.g., the so-called simplest cubic and quartic fields associated with the polynomials and . We give explicit formulas for powers of the Gaussian sums attached to the characters associated with these simplest number fields. We deduce a method for computing the exact values of these Gaussian sums. These values are then used to efficiently compute class numbers of simplest fields. Finally, such class number computations yield many examples of real cyclotomic fields of prime conductors and class numbers greater than or equal to . However, in accordance with Vandiver's conjecture, we found no example of for which divides .

     

Demjanenko matrix and recursion formula for relative class number over function fields

In this paper, we define Demjanenko matrix in function field and express the relative ideal class numbers as the determinant of this matrix. We also define another matrix which give us a recursion formula for the relative divisor class numbers h⁻(K_Pⁿ).     

Imaginary quadratic function fields

In Bautista-Ancona and Diaz-Vargas (2006) [B-D] a characterization and complete listing is given of the imaginary quadratic extensions K of k(x), where k is a finite field, in which the ideal class group has exponent two and the infinite prime of k(x) ramifies. The objective of this work is to give a characterization and list of these kind of extensions but now considering the case in which the infinite prime of k(x) is inert in K. Thus, we get all the imaginary quadratic extensions of k(x), in which the ideal class group has exponent two.