Quartic Fields Research Articles

Q-curves of degree 5 and jacobian surfaces of GL 2 -type

We construct a parametric family {E(±)(s,t,u)} of minimal Q-curves of degree 5 over the quadratic fields Q\(\), and the family {C(s,t,u)} of genus two curves over Q covering E{(+)(s,t,u) whose jacobians are abelian surfaces of GL2-type. We also discuss the modularity for them and the sign change between E{(+)(s,t,u) and its twist E(−)(s,t,u), which correspond by modularity to cusp forms of trivial and non-trivial Neben type characters, respectively. We find in {C(s,t,u)} concrete equations of curves over Q whose jacobians are isogenous over cyclic quartic fields to Shimura's abelian surfaces Af attached to cusp forms of Neben type character of level N= 29, 229, 349, 461, and 509.

On Two-parametric Quartic Families of Diophantine Problems

Two-parametric quartic Thue equations are completely solved for sufficiently large values of the parameters. Exceptional units are computed in related quartic number fields. The method depends heavily on A. Baker's theory of linear forms in logarithms and symbolic computation in MAPLE.

Formulae of type Ankeny-Artin-Chowla for class numbers of general cyclic sextic fields

LetK 6 be a real cyclic sextic number fields, andK 2, K3 be its quadratic and cubic subfields. Leth(L) denote the ideal class number of fieldL. Seven congruences forh =h(K6)/ h(K2)h(K3) are obtained. In particular, when conductorf 6 ofK 6 is primep, then $$Ch^ - \equiv B\frac{{p - 1}}{6}B\frac{{5\left( {p - 1} \right)}}{6}\left( {mod p} \right)$$ whereC is an explicitly given constant, andB n is the Bernoulli number. These results for real cyclic sextic fields are an extension of results for quadratic and cyclic quartic fields obtained by Ankeny-Artin-Chowla. Kiselev. Carlitz. Lu Hongwen, Zhang Xianke from 1948 to 1988.

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.

On Torsion Subgroups of Elliptic Curves with Integral $j$-Invariant over Imaginary Cyclic Quartic Fields

On Torsion Subgroups of Elliptic Curves with Integral $j$-Invariant over Imaginary Cyclic Quartic Fields

The conductor of a cyclic quartic field using Gauss sums

Let Q denote the field of rational numbers. Let K be a cyclic quartic extension of Q. It is known that there are unique integers A, B, C, D such that \(K = Q\left( {\sqrt {A(D + B\sqrt D )} } \right)\) where A is squarefree and odd, D=B2+C2 is squarefree, B\( > \) 0 , C\( > \) 0, GCD(A,D)=1. The conductor f(K) of K is f(K) = 2l|A|D, where \(l = \left\{ \begin{gathered} 3,{\text{ if }}D \equiv 2{\text{ }}({\text{mod 4}}){\text{ or }}D \equiv 1{\text{ (mod 4), }}B \equiv 1{\text{ }}({\text{mod 2}}), \hfill 2,{\text{ if }}D \equiv 1{\text{ (mod 4), }}B \equiv 0{\text{ (mod 2), }}A + B \equiv 3{\text{ (mod 4),}} \hfill 0,{\text{ if }}D \equiv 1{\text{ (mod 4), }}B \equiv 0{\text{ (mod 2), }}A + B \equiv 1{\text{ (mod 4)}}{\text{.}} \hfill \end{gathered} \right.\) A simple proof of this formula for f(K) is given, which uses the basic properties of quartic Gauss sums.

Computing with Subfields

The algorithms presented here make use of subfield information to improve computations. For the three problems we deal with in this paper, the solution of relative norm equations, computation of independent units in the Galois closure of quartic fields and the computation of relative integral bases in quadratic extensions of number fields, we will outline the necessary algorithms that use subfield information and whenever possible we will compare the running time of each algorithm with the equivalent method not using subfield information.

Hilbert Modular Varieties of Galois Quartic Fields

Letkbe a totally real number field of degreenoverQwith ring of integers Ok. The Hilbert modular variety ofkis a desingularization of the (natural) compactification of PSL2(Ok)\\Hn. This paper focuses on Hilbert modular varieties of Galois quartic fields containing units of norm −1. The main result is that each of these varieties has arithmetic genus greater than one and hence is not rational.

Zeros of Dedekind zeta functions in the critical strip

In this paper, we describe a computation which established the GRH to height 92 92 (resp. 40 40 ) for cubic number fields (resp. quartic number fields) with small discriminant. We use a method due to E. Friedman for computing values of Dedekind zeta functions, we take care of accumulated roundoff error to obtain results which are mathematically rigorous, and we generalize Turing’s criterion to prove that there is no zero off the critical line. We finally give results concerning the GRH for cubic and quartic fields, tables of low zeros for number fields of degree 5 5 and 6 6 , and statistics about the smallest zero of a number field.

Corrigendum Parity of class numbers and Witt equivalence of quartic fields

Proof. Take a nondyadic prime q of F with s(Fq) = 2 (there must be one since s(F ) = 2). We show that the order d of the ideal class [q] in the ideal class group of F is even. For suppose d is odd. Then q = (a) is a principal ideal and the number a is a q-adic prime times a q-adic square. Since −1 is not a local square at q, we have (−1, a)q = −1. On the other hand, we claim that (−1, a)p = 1 for all the remaining primes p of F. For a dyadic prime p this is obvious since −1 is a local square at p (the local dyadic levels are all equal to 1). When p is a finite nondyadic prime and p 6= q, then a is a local unit at p (by our choice of a), hence again the Hilbert symbol is trivial. Since s(F ) = 2, there are no real infinite primes, and at complex infinite primes any Hilbert symbol is trivial. This proves our claim, contradicting Hilbert reciprocity law. Hence d must be even, and so also the ideal class number of F is even. If K is any number field Witt equivalent to F, then K and F have the same Witt equivalence invariant. Hence, if F satisfies Conner’s level conditions, so does K, and, as has been already proved, it has even class number.

Discriminant of the normal closure of a dihedral quartic field

where s denotes the squarefree part of a 2 b 2 c, and the value of e ( = 2, 0, 2, 4, 6, 8) is given in Tables A (c -= 2 (mod 4)), B (c = 3 (mod 4)), C (c -= 5 (mod 8)), and D (c (mod 8)) of [3]. Tables A, B, C, D comprise 8, 8, 8, 27 cases respectively. Each case is specified by congruence conditions on a, b, and c. For example B 4 is the case a ~ b = 1 (mod 2), c -= 3 (rood 4), and in this case e = 8. In this paper we obtain the following formula for the discriminant of L.

Relative integral bases for quartic fields over quadratic subfields

Let L be a quartic number field with a quadratic subfield K. In 1986 Kawamoto gave a necessary and sufficient condition for L to have a normal relative integral basis (NRIB) over K. In this paper the authors explicitly construct a NRIB for L/K when such exists using their previous work on relative integral bases. The special cases when L is bicyclic, cyclic and pure are examined in detail.

Certain Quartic Fields with Small Regulators

For quartic fields with a quadratic subfield, explicit lower bounds of the regulators are given in terms of the discriminant. Further, these bounds are shown to be in a sense best possible by constructing infinitely many quartic fields with explicit fundamental units whose regulators approach our lower bounds as their discriminants become large. The fields constructed are new families of non-galois fields given by explicit quartic polynomials.

Normal relative integral bases for quartic fields over quadratic subfields

Let L be a quartic number field with a quadratic subfield K. In 1986 Kawamoto gave a necessary and sufficient condition for L to have a normal relative integral basis (NRIB) over K. In this paper the authors explicitly construct a NRIB for L/K when such exists using their previous work on relative integral bases. The special cases when L is bicyclic, cyclic and pure are examined in detail.

Orders of a quartic field

Orders of a quartic field

The conductor of a cyclic quartic field

The conductor of a cyclic quartic field

Parity of Class Numbers and Witt Equivalence of Quartic Fields

We show that 27 out of the 29 Witt equivalence classes of quartic number fields can be represented by fields of class number 1. It is known that the remaining two classes contain solely fields of even class numbers. We show that these two classes can be represented by fields of class number 2.

Integral Bases for Quartic Fields with Quadratic Subfields

Let L be a quartic number field with quadratic subfield Q([formula]). Then L = Q([formula], [formula]), where a + b[formula] is not a square in Q([formula]) and where a, b, and c may be taken to be integers with both c and ( a, b) squarefree. The discriminant of L, as well as an integral basis for L, is determined explicitly in terms of congruences involving a, b, and c. These results unify the existing results in the literature for quartic fields which are pure, bicyclic, cyclic, or dihedral, and complete the incomplete results in the literature for dihedral quartic fields. It is also shown that for each squarefree integer c there are infinitely many non- pure, dihedral quartic fields L with a power basis.

Parity of class numbers and Witt equivalence of quartic fields

We show that 27 out of the 29 Witt equivalence classes of quartic number fields can be represented by fields of class number 1. It is known that the remaining two classes contain solely fields of even class numbers. We show that these two classes can be represented by fields of class number 2.

The Exponent 2-Class-Group Problem for Non-Galois Over Q Quartic Fields That Are Quadratic Extensions of Imaginary Quadratic Fields

Focusing on a particular case, we will show that one can explicitly determine the quartic fields K that have ideal class groups of exponent ≤ 2, provided that K/ Q is not normal, provided that K is a quadratic extension of a fixed imaginary quadratic number field, and provided that the regulator of K is not too large compared with the discriminant of K.