Imaginary Abelian Number Fields Research Articles

Annihilators of the Ideal Class Group of an Imaginary Abelian Number Field

Annihilators of the Ideal Class Group of an Imaginary Abelian Number Field

The imaginary abelian number fields of $2$-power degrees with ideal class groups of exponent $≤2$

The imaginary abelian number fields of $2$-power degrees with ideal class groups of exponent $≤2$

A generalization of Newman’s formula

We give a determinant formula for the relative class number of an imaginary abelian number field, which is a generalization of Newman’s formula of 1970 and of Skula’s of 1981. By our formula we can determine the signs of these determinants, which these authors did not give.

Lower bounds for relative class numbers of imaginary abelian number fields and CM-fields

We give explicit upper and lower bounds for relative class numbers of imaginary abelian number fields and of nonabelian CM-fields K. We then use them to obtain Brauer–Siegel type results for relative class numbers of CM-fields. The main feature of this paper is a new method which enables us to easily deal with the case where K contains an imaginary quadratic subfield. We will make use of the ideas introduced in [Lou04] to obtain in Theorem 1 upper bounds on |L(1, χ)| for even and odd primitive Dirichlet characters χ, whereas our previous ideas (developed in [Lou98b] and [Lou01]) failed to embrace the case of odd characters. These bounds depend on whether L(s, χ) has or does not have a real zero β in the range 0 < β < 1. They will then enable us to obtain in Theorems 18, 28 and 31 bounds for relative class numbers hK of CM-fields K, especially in the case that K contains an imaginary quadratic subfield L. Apart from the proof of Lemma 17, which can be found in [Lou03], this paper provides the reader with a self-contained exposition of how one can obtain (as in Corollaries 20, 21, 23 and 25 where several footnotes clearly show that our approach is more efficient than the ones formerly developed by various authors) good enough explicit lower bounds for relative class numbers to enable him to solve various class number problems for CM-fields or to simplify the existing proofs (e.g., see [CK98], [CK00a], [CK00b], [Lou95], [Lou97],[Lou98a], [Lou99], [MM] and [Yam]). Whereas almost all the papers in the literature dealing with explicit lower bounds for relative class numbers of CM-fields (or values at s = 1 of L-functions)

Generalizations of girstmair’s formulas

In 1994 and 1995 GIRSTMAIR gave (relative) class number formulas for the imaginary quadratic field $$\mathbb{Q}(\sqrt { - p} )$$ , P an odd prime with p ≡ 3 (mod 4) and p ≥ 7, using the coefficients of the digit expression of 1/p and z/p, respectively, where z is an integer with 1 ≤ z ≤p - 1. We extend the formulas to an imaginary abelian number field.

The class groups of the imaginary Abelian number fields with Galois group (ℤ/2ℤ)n

Assuming the Generalised Riemann Hypothesis we determine all imaginary Abelian number fieldsNwhose Galois groupG(N/ℚ) is isomorphic to (ℤ/2ℤ)nfor some integersn≥ 1 and the square of every ideal ofNis principal.

New results on the nonexistence of generalized bent functions

Several new results on the nonexistence of generalized bent functions are proved by using properties of the decomposition law of primes and the class group of imaginary Abelian number fields.

A relative class number formula for an imaginary abelian number field by means of Dedekind sum

We give a relative class number formula for an imaginary abelian number field by means of some ``Dedekind sum''. This is a generalization of Carlitz and Olson's formula presented in 1955.

Inkeri‚s determinant for an imaginary abelian number field

We give a relative class number formula for an imaginary abelian number field by means of a determinant, which yields a generalization of Inkeri‚s determinant.

New results on non-existence of generalized bent functions (II)

Several new results on non-existence of generalized bent functions are presented by using the class group of related imaginary abelian number fields.

A generalization of maillet and demyanenko determinants for the Cyclotomic Zp-Extension

LetK be an imaginary abelian number field. By means of a generalization of Maillet and Demyanenko determinants we give a relative class number formula for an intermediate field of the cyclotomic ℤp-extension ofK. The degree of the generalized determinant is a half of the degree ofK over ℚ.

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.

Class numbers of imaginary abelian number fields

Let N be an imaginary abelian number field. We know that h-, the relative class number of N, goes to infinity as fN, the conductor of N, approaches infinity, so that there are only finitely many imaginary abelian number fields with given relative class number. First of all, we have found all imaginary abelian number fields with relative class number one: there are exactly 302 such fields. It is known that there are only finitely many CMfields N with cyclic ideal class groups of 2-power orders such that the complex conjugation is the square of some automorphism of N. Second, we have proved in this paper that there are exactly 48 such fields.

The imaginary abelian number fields with class numbers equal to their genus class numbers

Titre français : Sur les corps abéliens dont le nombre de classes est égal au nombre de genres. Nous savons qu'il n'existe qu'un nombre fini de corps abéliens imaginaires pour lesquels le nombre de classes est égal au nombre de genres. Ceux de ces corps qui sont cycliques et non quadratiques ont été classés dans [Lou2,4] et dans [CK]. Dans cet article, nous déterminons tous les corps abéliens non cycliques dont le nombre de classes est égal au nombre de genres. Cela achève la classification des corps abéliens possédant une classe par genre, sauf dans le cas des corps quadratiques imaginaires.

The nonquadratic imaginary cyclic fields of 2-power degrees with class numbers equal to their genus class numbers

It is known that there are only finitely many imaginary abelian number fields with class numbers equal to their genus class numbers. Here, we determine all the imaginary cyclic fields of 2-power degrees with class numbers equal to their genus class numbers.

The imaginary cyclic sextic fields with class numbers equal to their genus class numbers

It is known that there are only finitely many imaginary abelian number fields with class numbers equal to their genus class numbers. Here, we determine all the imaginary cyclic sextic fields with class numbers equal to their genus class numbers.

Computation of Relative Class Numbers of Imaginary Abelian Number Fields

We develop an efficient technique for computing relative class numbers of imaginary abelian fields, efficient enough to enable us to easily compute relative class numbers of imaginary cyclic fields of degrees 32 and conductors greater than 1013, or of degrees 4 and conductors greater than 1015. Acccording to our extensive computation, all the 166204 imaginary cyclic quartic fields of prime conductors p less than 107 have relative class numbers less than p/2. Our major innovation is a technique for computing numerically root numbers appearing in some functional equations.

A generalization of Maillet and Demyanenko determinants

where the product ∏ χ is taken over the odd primitive characters χ of K with conductor f(χ) (see Hasse [4]). Several ways of representing the product in (1) by a determinant are known, some of them holding for certain types of fields only (see Carlitz and Olson [1], Hazama [5] and further references in Hirabayashi [6]). Here our main concern is a unified approach to the Maillet determinant on the one hand and to the Demyanenko determinant on the other hand. This approach relies on the “b-division vector” introduced by Girstmair [3]. We obtain a relative class number formula for an arbitrary imaginary abelian number field which generalizes formulae of Girstmair [2] and [6]. Tsumura [7] also generalized both type of determinant formulae, but his generalization for the Demyanenko determinant requires the oddness of the conductor.

A finiteness theorem for imaginary abelian number fields

Lately, I. Miyada proved that there are only finitely many imaginary abelian number fields with Galois groups of exponents ≤2 with one class in each genus. He also proved that under the assumption of the Riemann hypothesis there are exactly 301 such number fields. Here, we prove the following finiteness theorem: there are only finitely many imaginary abelian number fields with one class in each genus. We note that our proof would make it possible to find an explict upper bound on the discriminants of these number fields which are neither quadratic nor biquadratic bicyclic. However, we do not go into any explicit determination.

On Bases of Groups of Circular Units of Some Imaginary Abelian Number Fields

Letkbe an imaginary abelian number field whose conductor has at most two distinct prime divisors and whose maximal subfields which are ramified at one prime are imaginary. We shall find a basis for the groupCof circular units inkand compute the index ofCin the groupEof units ink.