Arithmetik Abelscher Varietäten mit komplexer Multiplikation (Q1059687)

This interesting work is, roughly speaking, on the investigation of the interaction of the arithmetic of CM-fields and the geometry of CM-varieties, through Grössencharacters of type \((A_ 0)\). One of the major topics concerns with the conditions under which the Grössencharacter gives the annihilation of the class group of the CM-field. The paper includes a summary written very well in German with its English version. So the reviewer recommends everyone to read it and here only mentions few points. The first chapter is titled by ``Grössencharaktere vom Typ \(A_ 0\)''. Here the author introduces the so-called test-character whose existence is ensured in ``Schlüssel-Lemma'' at page 12 and gives Iwasawa's theorem on annihilation of class group. Also Jacobian sums are considered from the view point of Grössencharacter of type \(A_ 0\). The title of the second chapter is ``Abelsche Varietäten mit komplexer Multiplikation'' in which are treated the Grössencharacter of CM-variety and the CM-variety of Grössencharacter. The third chapter is on abstract theory of abelian CM-types and includes some numerical computations for the rank of the half-system of abelian CM-type and index formula. ``Geometrische Annulator-Kriterien'' is the title of the fourth chapter and ``Modulikörper und unverzweigte Erweiterungen'', the fifth, includes the generalization of the notion of \(\mathbb Q\)-curves introduced by Gross to \(\mathbb Q\)-varieties with the help of test-character and the field of moduli. In the last chapter ``Die CM-Varietäten der Fermat-Jacobischen'', the author describes \(\mathbb Q\) isogenous splitting of the Jacobian of the Fermat-curve into CM-varieties and with the aid of Jacobi sums as the corresponding Grössencharacter, he obtains a proof of Kummer-Jacobi relations without Gauss sums.