Franz Lemmermeyer

Ideal class groups of cyclotomic number fields III

Following Hasses example, various authors have been deriving divisibility properties of minus class numbers of cyclotomic fields by carefully examining the analytic class number formula. In this paper we will show how to generalize these results to CM-fields by using class field theory. Although we will only need some special cases, we have also decided to include a few results on Hasses unit index for CM-fields as well, because it seems that our proofs are more direct than those given by Hasse.

Imaginary quadratic fields with Cl2(k)≃(2,2,2)

Abstract We characterize those imaginary quadratic number fields, k, with 2-class group of type (2,2,2) and with the 2-rank of the class group of its Hilbert 2-class field equal to 2. We then compute the length of the 2-class field tower of k.

Galois action on class groups

Abstract It is well known that the Galois group of an extension L/F puts constraints on the structure of the relative ideal class group Cl(L/F). Explicit results, however, hardly ever go beyond the semisimple abelian case, where L/F is abelian (in general cyclic) and where (L:F) and #Cl(L/F) are coprime. Using only basic parts of the theory of group representations, we give a unified approach to these as well as more general results.

The Development of the Principal Genus Theorem

Genus theory belongs to algebraic number theory and, in very broad terms, deals with the part of the ideal class group of a number field that is ‘easy to compute’. Historically, the importance of genus theory stems from the fact that it was the essential algebraic ingredient in the derivation of the classical reciprocity laws – from Gaus’s second proof over Kummer’s contributions up to Takagi’s ‘general’ reciprocity law for p-th power residues. The central theorem in genus theory is the principal genus theorem, which is hard to describe in just one sentence – readers not familiar with genus theory might want to glance into Section 2 before reading on. In modern terms, the principal genus theorem for abelian extensions k/Q describes the splitting of prime ideals of k in the genus field kgen of k, which by definition is the maximal unramified extension of k that is abelian over Q. In this note we outline the development of the principal genus theorem from its conception in the context of binary quadratic forms by Gaus (with hindsight, traces of genus theory can be found in the work of Euler on quadratic forms and idoneal numbers) to its modern formulation within the framework of class field theory. It is somewhat remarkable that, although the theorem itself is classical, the name ‘principal ideal theorem’ (‘Hauptgeschlechtssatz’ in German) was not used in the 19th century, and it seems that it was coined by Hasse in his Bericht [28] and adopted immediately by the abstract algebra group around Noether. It is even more remarkable that Gaus doesn’t bother to formulate the principal genus theorem except in passing: after observing in [25, §247] that duplicated classes (classes of forms composed with themselves) lie in the principal genus, the converse (namely the principal genus theorem) is stated for the first time in §261: si itaque omnes classes generis principalis ex duplicatione alicuius classis provenire possunt (quod revera semper locum habere in sequentibus demonstrabitur), . . . 1 The actual statement of the principal genus theorem is somewhat hidden in [25, §286], where Gaus formulates the following Problem. Given a binary form F = (A,B,C) of determinant D belonging to a principal genus: to find a binary form f from whose duplication we get the form F . It strikes us as odd that Gaus didn’t formulate this central result properly; yet he knew exactly what he was doing [25, §287]:

Some families of non-congruent numbers

Let E_k denote the elliptic curve defined by y^2 = x(x^2 - k^2). We consider the curves with k = pl, p = l = 1 mod 8 primes, and show that the density of rank-0 curves among them is at least 1/2 by explicitly constructing nontrivial elements in the 2-part of the Tate-Shafarevich group of E_k.

Counterexamples to the Hasse Principle

Abstract This article explains the Hasse principle and gives a self-contained development of certain counterexamples to this principle. The counterexamples considered are similar to the earliest counterexample discovered by Lind and Reichardt. This type of counterexample is important in the theory of elliptic curves: today they are interpreted as nontrivial elements in Tate–Shafarevich groups.

Parametrization of Algebraic Curves from a Number Theorist's Point of View

Abstract We present the technique of parametrization of plane algebraic curves from a number theorists point of view and present Kapferers simple and beautiful (but little known) proof that nonsingular curves of degree > 2 cannot be parametrized by rational functions.

Selmer groups and quadratic reciprocity

In this article we study the 2-Selmer groups of number fieldsF as well as some related groups, and present connections to the quadratic reciprocity law inF.

RELATIONS IN THE 2-CLASS GROUP OF QUADRATIC NUMBER FIELDS

We construct a family of ideals representing ideal classes of order 2 in quadratic number fields and show that relations between their ideal classes are governed by certain cyclic quartic extensions of the rationals.

Die Pellsche Gleichung

In diesem Kapitel werden wir zeigen, dass die Pellsche Gleichung \(x^{2}-my^{2}=1\) fur alle positiven Nichtquadratzahlen m eine ganzzahlige Losung mit y ≠ 0 besitzt. Dies bedeutet, dass jeder reellquadratische Zahlring nichttriviale Einheiten besitzt, die das Rechnen in solchen Zahlkorpern je nach standpunkt komplizieren oder interessant machen. Zur Berechnung der Idealklassengruppe eines reellquadratischen Zahlkorpers ist die Kenntnis einer Losung der entsprechenden Pellschen Gleichung unverzichtbar.