Quartic Number Fields Research Articles

The class number one problem for some totally complex quartic number fields

We prove that there are 95 non-isomorphic totally complex quartic fields whose rings of algebraic integers are generated by an algebraic unit and whose class numbers are equal to 1. Moreover, we prove Louboutin's Conjecture according to which a totally complex quartic unit ε u generally generates the unit group of the quartic order Z [ ε u ] .

Reflection Principles and Bounds for Class Group Torsion

We introduce a new method to bound -torsion in class groups, combining analytic ideas with reflection principles. This gives, in particular, new bounds for the 3-torsion part of class groups in quadratic, cubic and quartic number fields, as well as bounds for certain families of higher degree fields and for higher . Conditionally on GRH, we obtain a nontrivial bound for -torsion in the class group of a general number field.

ON THE TORSION OF ELLIPTIC CURVES OVER QUARTIC NUMBER FIELDS

We determine which groups ${\mathbb{Z}}/M{\mathbb{Z}}\oplus{\mathbb{Z}}/N{\mathbb{Z}}$ occur infinitely often as torsion groups $E(K)_{\operatorname{tors}}$ when $K$ varies over all quartic number fields and $E$ varies over all elliptic curves over $K$ .

Computing the arithmetic genus of Hilbert modular fourfolds

The Hilbert modular fourfold determined by the totally real quartic number field k is a desingularization of a natural compactification of the quotient space Γ k \H 4 , where Γ k = PSL 2 (O k ) acts on H 4 by fractional linear transformations via the four embeddings of k into R. The arithmetic genus, equal to one plus the dimension of the space of Hilbert modular cusp forms of weight (2,2,2,2), is a birational invariant useful in the classification of these varieties. In this work, we describe an algorithm allowing for the automated computation of the arithmetic genus and give sample results.

THE BRAUER–SIEGEL THEOREM

Explicit bounds are given for the residues at of the Dedekind zeta functions of number fields. As a consequence, a simple proof of the Brauer-Siegel theorem and explicit lower bounds for class numbers of number fields are obtained. Compared with Stark's original approach, the paper is explicit and more satisfactory for number fields containing quadratic subfields. Examples are given of fully explicit lower bounds for class numbers of various types of number fields, for example normal and non-normal number fields of odd degree, with an emphasis on cubic fields, real cyclic quartic number fields, and non-normal quartic number fields containing an imaginary quadratic subfield.

The simplest quartic fields with ideal class groups of exponents less than or equal to 2

The simplest quartic fields are the real cyclic quartic number fields defined by the irreducible quartic polynomials x4-mx3-6x2+mx+1, where m runs over the positive rational integers such that the odd part of m2+16 is squarefree. We give an explicit lower bound for their class numbers which is much better than the previous known ones obtained by A. Lazarus. Then, using it, we determine the simplest quartic fields with ideal class groups of exponents ≤2.

Determination of All [formula omitted]-Rational CM-Points in the Moduli Space of Principally Polarized Abelian Surfaces

It is known that in the moduli space A1 of elliptic curves, there exist precisely 9 Q-rational points corresponding to the isomorphism class of elliptic curves with complex multiplication by the ring of algebraic integers of a principal imaginary quadratic number field. Here, we prove that in the moduli space A2 of principally polarized abelian surfaces, there exist precisely 19 Q-rational points corresponding to the isomorphism class of abelian surfaces whose endomorphism rings are isomorphic to the rings of algebraic integers of some imaginary cyclic quartic number fields.

On the solutions of a family of quartic Thue equations

In this paper, we solve a certain family of diophantine equations associated with a family of cyclic quartic number fields. In fact, we prove that for n < 5 × 10 6 and n ≥ N = 1.191 x 10 19 , with n, n + 2, n 2 + 4 square-free, the Thue equation Φ n (x,y) = x 4 - n 2 x 3 y - (n 3 + 2n 2 + 4n + 2)x 2 y 2 - n 2 xy 3 + y 4 = 1 has no integral solution except the trivial ones: (1, 0), (-1,0), (0,1), (0, -1).

Power Bases in Dihedral Quartic Fields

A quartic number field,L, is calleddihedralif the normal closure,N, ofLsatisfies Gal(N/Q)≅D8. We investigate whether or not the ring of integers of such a quartic field has a power basis. When the quadratic subfield ofLis imaginary, the problem is completely solved. When it is real, the same method leads to a solution in many cases. Several numerical illustrations of the method are given.

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.

Exceptional units in a family of quartic number fields

We determine all exceptional units among the elements of certain groups of units in quartic number fields. These groups arise from a one-parameter family of polynomials with two real roots.

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.

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.

Simultaneous Representation of Integers by a Pair of Ternary Quadratic Forms—With an Application to Index Form Equations in Quartic Number Fields

LetQ1, Q2∈Z[X, Y, Z] be two ternary quadratic forms andu1, u2∈Z. In this paper we consider the problem of solving the system of equations[formula]According to Mordell [12] the coprime solutions of[formula]can be presented by finitely many expressions of the formx=fx(p, q),y=fy(p, q),z=fz(p, q), wherefx, fy, fz∈Z[P, Q] are binary quadratic forms andp, qare coprime integers. Substituting these expressions into one of the equations of (1), we obtain a quartic homogeneous equation in two variables. If it is irreducible it is a quartic Thue equation, otherwise it can be solved even easier. The finitely many solutionsp, qof that equation then yield all solutionsx, y, zof (1). We also discuss two applications. In [8] we showed that the problem of solving index form equations in quartic numbers fieldsKcan be reduced to the resolution of a cubic equationF(u, v)=iand a corresponding system of quadratic equationsQ1(x, y, z)=u,Q2(x, y, z)=v, whereFis a binary cubic form andQ1, Q2are ternary quadratic forms. In this case the application of the new ideas for the resolution of (1) leads to quartic Thue equations which split over the same quartic fieldK. The second application is to the calculation of all integral points of an elliptic curve.

On a quartic diophantine equation

In this paper we consider the quartic diophantine equation 3(y2 – 1) = 2x2(x2 – 1) in integers x and y. We show that this equation does not have any other solutions (x, y) with x≧0 than those given by x = 0,1,2,3,6,91. Two approaches are emphasized, one based on diophantine approximation techniques, the other depends on the structure of certain quartic number fields.

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.

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.

Determining the Monogeneity of a Quartic Number Field

AbstractThis note presents a method that determines all power integral bases of a quartic number field by solving Thue equations of degrees 3 and 4. To this end, projective representations of the ring of integers by graded complete intersections are studied and a criterion for monogeneity in terms of projective representations is derived.

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.