Complex Quadratic Fields Research Articles

Weber's class invariants

Weber proves in §§114-124 of his Algebra [19] that if w is complex quadratic and ℤ[ω] is the ring of integers of the field ℚ(ω) then the absolute class field of ℚ(ω) is generated by the modular invariant j(w); he calls j(w) a class invariant. He goes on in §§125-144 to consider the values f(w) of other modular functions f(z); he shows that in certain cases the degree of the extension ℚ(ω, f(ω)) of ℚ(ω, j(ω)) is much less than that of ℚ(f(z)) over ℚ(f(z)); indeed, f(ω) is often in ℚ(ω, j(ω)), and in such circumstances Weber calls f(ω) a class invariant too. Using such results, Weber computes many class invariants—an end in itself, since the numbers are so beautiful. More recently, results of this type have been applied to determine all the complex quadratic fields with class number 1, and to prove that elliptic curves of certain families always have infinitely many rational points-see [9, 2, 5].

Shorter Notes: A Historical Note on Complex Quadratic Fields with Class- Number One

Shorter Notes: A Historical Note on Complex Quadratic Fields with Class- Number One

The role of modular functions in a class-number problem

The author has previously shown that there are exactly nine complex quadratic fields of class-number one. Here we show that the proof rests on some extremely interesting identities in modular functions which also provide a connection with the work of Heegner.

A historical note on complex quadratic fields with class-number one.

Let d be the discriminant of a quadratic field, h(d) the class-number of the field and Xd the character (mod I dj) given by Xd(n) = (d/n) (Kronecker symbol). If we restrict d to values <-8 with h(d) = 1 then I dj is a prime_3 (mod 8) and it has recently been proved [I], [4] that d ? 163. The purpose of this note is to show that Gelfond and Linnik missed an opportunity to prove this result in 1949. Let k be the discriminant of a real quadratic field and Ec the fundamental unit of Q(x/k). In the sequel, we assume d < -8 and h(d) = 1. The following expansion was known for prime k by Heilbronn and Linfoot [31 and has recently been established [5] for all k such that (k, d) = 1,

On the “gap” in a theorem of Heegner

In 1952, Kurt Heegner gave a proof of the fact that there are exactly nine complex quadratic fields of class-number one. His proof rests on the fact that a certain 24th degree polynomial with rational coefficients has a 6th degree factor which also has rational coefficients. Unfortunately, this reducibility has never been justified. In this paper, we fill this gap in Heegner's proof.

A complete determination of the complex quadratic fields of class-number one.

A complete determination of the complex quadratic fields of class-number one.

On Complex Quadratic Fields with Class Number Equal to One

On Complex Quadratic Fields with Class Number Equal to One

On complex quadratic fields with class number equal to one

Let R(V p) be a quadratic extension of the rationals, where p is a positive square free integer. For nine values of p, namely 1, 2, 3, 7, 11, 19, 43, 67, 163, the integers of R(/ p) form a unique factorization domain. Heilbronn and Linfoot [1] have shown that there is at most one more such value of p, and Lehmer [2] has shown that p must be a prime greater than 5 * 109. In the present paper we verify and extend the lower bound of 5 * 109 for p. The result is

The geometry of numbers over algebraic number fields

1. The Geometry of Numbers was founded by Minkowski in order to attack certain arithmetical problems, and is normally concerned with lattices over the rational integers. Minkowski himself, however, also treated a special problem over complex quadratic number fields [5], and a number of writers have since followed him. They were largely concerned with those fields which have class-number h= 1; and this simplification removes many of the characteristic features of the more general case. Hermann Weyl [10] gave a thorough account of the extension of Minkowski's theory of the reduction of quadratic forms to gauge functions over general algebraic number fields and quaternion algebras, and we shall follow part of his developments, though our definition of a lattice is quite different. The desirability of extending the Geometry of Numbers to general algebraic number fields was emphasized by Mahler in a seminar at Princeton. In this paper we shall carry out this program, extending the fundamental results of Mahler [4] to our more general case and applying them to specific problems. Certain new ideas are necessary, but much of this paper must be regarded as expository. In particular, when the proof of a result is essentially analogous to that for the real case we have merely given a reference to the latter. 2. Let K be an algebraic extension of the rationals of degree m. We regard K as an algebra over the rationals, which we can extend to an algebra K* over the reals. It is well known that K* is commutative and semi-simple (being in fact isomorphic to the direct sum of r copies of the reals and s copies of the complex numbers, where r and 2s are the number of real and complex conjugates of K); and the integers of K* are just those of K. We now define the n-dimensional space Kn over K as being the set of ordered n-tuples of elements in K*. Any tCK* is of the form t=x1i1+ * * * +xwmU, where the x, are real and co,, * * *, com is an integral basis for K; and hence there is a natural map of Kn onto Rmn in which each component t is mapped onto m of the components of the point in Rmn, namely xi, x, m as above. We can define a metric and a measure in Kn by means of those in Rmn, with the above map, and so Kn is a locally compact complete metric space.

Complex homogeneous linear forms

LetZ, Q, Cdenote respectively the ring of rational integers, the field of rational numbers and the field of complex numbers. Minkowski (4) solved the problem of minimizingforx, y ∈ Z(i)orZ(ρ), wherea, b, c, d ∈ Chave fixed determinant Δ ≠ 0. Here ρ = exp 2/3πi, andZ(i)andZ(p)are the rings of integers inQ(i)andQ(ρ)respectively. In fact he found the best possible resultsforZ(i), andforZ(ρ), wherewhile Buchner (1) used Minkowski's method to show thatforZ(i√2). Hlawka(3) has also proved (1·2), and Cassels, Ledermann and Mahler (2) have proved both (1·2) and (1·3). In a paper being prepared jointly by H. P. F. Swinnerton-Dyer and the author, general problems of the geometry of numbers in complex space are discussed and a systematic method given for solving the above problem for all complex quadratic fields Q(ϑ). Here, ϑ is a non-real number satisfying. an irreduc7ible quadratic equation with rational coefficients. The above problem is solved in detail forQ(i√5), for whichand the ‘critical forms’ can be reduced to