Exponents of the ideal class groups of CM number fields

Since class numbers of CM number fields of a given degree go to infinity with the absolute values of their discriminants, it is reasonable to ask whether the same conclusion still holds true for the exponents of their ideal class groups. We prove that under the assumption of the Generalized Riemann Hypothesis this is indeed the case. Received: 8 May 2001; in final form: 15 April 2002/Published online: 8 November 2002     

Class number and class group problems for some non-normal totally real cubic number fields

Let \lcub;K _m } _{m ≥ 4} be the family of non-normal totally real cubic number fields associated with the Q-irreducible cubic polynomials P _m (x) =x ³−mx ²−(m+1)x− 1, m≥ 4. We determine all these K _m 's with class numbers h _m ≤ 3: there are 14 such K _m 's. Assuming the Generalized Riemann hypothesis for all the real quadratic number fields, we also prove that the exponents e _m of the ideal class groups of these K _m go to infinity with m and we determine all these K _m 's with ideal class groups of exponents e _m ≤ 3: there are 6 suchK _m with ideal class groups of exponent 2, and 6 such K _m with ideal class groups of exponent 3. Received: 16 November 2000 / Revised version: 16 May 2001     

On class numbers of quadratic extensions¶over function fields

We consider class numbers of quadratic extensions over a fixed function field. We will show that there exist infinitely many quadratic extensions which have class numbers not being divisible by 3 and satisfy prescribed ramification conditions. Received: 24 October 1997 / Revised version: 26 February 1998     

On primitive roots of tori: The case of function fields

We generalize Bilharz's Theorem for to all one-dimensional tori over global function fields of finite constant field. As an application, we also derive an analogue, in the setting of function fields, of a theorem (Chen-Kitaoka-Yu, Roskam) on the distribution of fundamental units modulo primes. Received: 16 October 2000 / Published online: 2 December 2002 Research partially supported by National Science Council, Rep. of China.     

Higher relative class number formulae

Let E be a totally complex abelian number field with maximal real subfield F, and let denote the non-trivial character of . Similar to the classical case n=1 the value of the Artin L-function at for odd is given by a relative class number formula of the form Here is a higher Q-index, which is equal to 1 or 2 and is a higher relative class number. Here for any number field L the higher class number is the order of the finite group closely related to the order of the higher K-theory group of the ring of integers in L. Received: 4 June 1999 / Revised version: 27 September 2001 / Published online: 26 April 2002     

Théorie d'Iwasawa des corps p-rationnels et p-birationnels

We study Iwasawa theory for p-rational and p-birational fields. A classical invariant characterises them and, in the case of CM-fields, this gives an explicit characterisation. We show how to compute those fields and and give numerical examples for small degrees.

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Received: 20 May 1997 / Revised version: 9 April 1998     

Small generators of number fields

If K is a number field of degree n over Q with discriminant D _K and if α∈K generates K, i.e. K=Q(α), then the height of α satisfies with . The paper deals with the existence of small generators of number fields in this sense. We show: (1) For each $n$ there are infinitely many number fields K of degree $n$ with a generator α such that . (2) There is a constant d ² such that every imaginary quadratic number field has a generator α which satisfies .?(3) If K is a totally real number field of prime degree n then one can find an integral generator α with . Received: 10 January 1997 / Revised version: 13 January 1998     

Initial layers of Zp-extensions and Greenberg's conjecture

Let p be a prime number, K a finite abelian extension of Q containing p-th roots of unity and K _n the n-th layer of the cyclotomic Z _p -extension of K. Under some conditions we construct an element of K _n from an ideal class of the maximal real subfield of K _n . We determine whether its p-th root is contained by some Z _p -extension of K _n or not for each n, using the zero of p-adic L-function and the order of the ideal class group of the maximal real subfield of K _m for sufficiently large m. Received: 13 February 1998 / Revised version: 30 September 1998     

Finiteness of real quadratic fields which admit positive integral diagonal septanary universal forms

In this paper, we will prove if D is large enough, there are no positive integral diagonal septanary universal quadratic forms over . Received: 13 November 1997 / Revised version: 17 November 1998     

Two-primary algebraic K-theory of two-regular number fields

We explicitly calculate all the 2-primary higher algebraic K-groups of the rings of integers of all 2-regular quadratic number fields, cyclotomic number fields, or maximal real subfields of such. Here 2-regular means that (2) does not split in the number field, and its narrow Picard group is of odd order. Received August 1, 1998     

On Sums of Roots of Unity

 Suppose that for some root of unity ζ of order Q with and all coefficients a _i belonging to a number field L. We bound Q in terms of k and . This generalizes a result of Conway and Jones for the case of rational coefficients. Moreover, we give an application to linear relations among characteristic functions of arithmetical progressions. (Received 18 January 1999; in revised from 14 June 1999)     

On the Galois cohomology of ideal class groups

We use étale cohomology to prove some explicit results on the Galois cohomology of ideal class groups. Received: 3 May 2007     

On quadrature formulae with maximal trigonometric degree of precision

Summary. In [DD] the problem of existence and uniqueness of a quadrature formula (QF) with maximal trigonometric degree of precision (MTDP) with a fixed number of free nodes and fixed different multiplicities at each node is considered. Even the affirmative answer to the question of existence and uniqueness is useless from a practical point of view if the QF is not explicitly found or if a complete characterization for the nodes and for the coefficients of the QF is not given. On the other hand the problem of the complete constructive characterization of the QF with MTDP is one of the main problems in the theory of numerical integration. In this paper we give a complete constructive characterization for the QF with MTDP in the case of a special type of periodic multiplicities. The results can be considered as a natural generalization of the previous results, which are given in [GO] (one-periodic case of multiplicities) and [DD] (two-periodic case of multiplicities). We evaluate the practical usefulness of the optimal numerical methods, which are obtained. Received June 16, 1995 / Revised version received April 3, 1996     

On maximal curves in characteristic two

The genus g of an q-maximal curve satisfies g=g ₁≔q(q−1)/2 or . Previously, q-maximal curves with g=g ₁ or g=g ₂, q odd, have been characterized up to q-isomorphism. Here it is shown that an q-maximal curve with genus g ₂, q even, is q-isomorphic to the non-singular model of the plane curve ∑ _{i =1}} ^t y ^{q /2 i} =x ^{q +1}, q=2 ^t , provided that q/2 is a Weierstrass non-gap at some point of the curve. Received: 3 December 1998     

Théorie des genres des corps globaux

We establish the fundamental results of genus theory for finite (non necessary Galois) extensions of global fields by using narrow S-class groups, when S is an arbitrary finite set of places. This exposition, which involves both the number fields and the functions fields cases, generalizes most classical results on this subject. Received: 8 February 1999 / Revised version: 17 December 1999     

Note on g-Hilbertian fields and nonrational curves¶with the Hilbert property

In a recent paper, Fried and Jarden prove the existence, for all integers g, of non-Hilbertian fields K which cannot be covered by a finite number of sets of the form ϕ (X(K)), where X is a curve of genus ≤g and ϕ is a rational function on X of degree ≥ 2. (If no bound is given on the genus we recover the notion of Hilbertian field.) This generalizes the case g=0, obtained previously by Corvaja and Zannier with a more elementary method. By a suitable modification of that method, we give here a new proof of the result of Fried and Jarden which avoids the use of deep group theoretical results. By a somewhat related construction we give an example of a curve X/Q of any prescribed genus and a Hilbertian field K⊂ˉQ such that X/K has the Hilbert property, i.e. the set of rational points X(K) is not thin. Received: 10 March 1998 / Revised version: 20 April 1998