The class number one problem for the dihedral and dicyclic CM-fields
We recall the determination of all the dihedral CM-fields with relative class number one, and prove that dicyclic CM-fields have relative class numbers greater than one.
The class number one problem for the non-abelian normal CM-fields of degree 16
The class number one problem for the non-abelian normal CM-fields of degree 24 and 40
The class number one problem for the non-normal sextic CM-fields. Part 2
The class number one problem for the real quadratic fields ℚ(√((an)²+4a))
We solve unconditionally the class number one problem for the 2-parameter family of real quadratic fields ℚ(√d) with square-free discriminant d = (an)²+4a for positive odd integers a and n.
The Cohen-Lenstra heuristics, moments and -ranks of some groups
This article deals with the coherence of the model given by the Cohen-Lenstra heuristic philosophy for class groups and also for their generalizations to Tate-Shafarevich groups. More precisely, our first goal is to extend a previous result due to É. Fouvry and J. Klüners which proves that a conjecture provided by the Cohen-Lenstra philosophy implies another such conjecture. As a consequence of our work, we can deduce, for example, a conjecture for the probability laws of -ranks of Selmer groups...
The complete determination of wide Richaud-Degert types which are not 5 modulo 8 with class number one
The digits of 1/p in connection with class number factors
The Diophantine equation
The distribution of second -class groups on coclass graphs
General concepts and strategies are developed for identifying the isomorphism type of the second -class group , that is the Galois group of the second Hilbert -class field , of a number field , for a prime . The isomorphism type determines the position of on one of the coclass graphs , , in the sense of Eick, Leedham-Green, and Newman. It is shown that, for special types of the base field and of its -class group , the position of is restricted to certain admissible branches of coclass...
The field descent and class groups of CM-fields
The Heegner-Stark-Baker-Deuring-Siegel Theorem.
The homology of cyclic branched covers of S3.
The homology of irregular dihedral branched covers of S3.
The imaginary abelian number fields with class numbers equal to their genus class numbers
We know that there exist only finitely many imaginary abelian number fields with class numbers equal to their genus class numbers. Such non-quadratic cyclic number fields are completely determined in [Lou2,4] and [CK]. In this paper we determine all non-cyclic abelian number fields with class numbers equal to their genus class numbers, thus the one class in each genus problem is solved, except for the imaginary quadratic number fields.
The imaginary cyclic sextic fields with class numbers equal to their genus class numbers
It is known that there are only finitely many imaginary abelian number fields with class numbers equal to their genus class numbers. Here, we determine all the imaginary cyclic sextic fields with class numbers equal to their genus class numbers.
The imaginary quadratic fields of class number 4
The Iwasawa invariants and the higher -groups associated to real quadratic fields.
The Lifted Root Number Conjecture for some cyclic extensions of ℚ
The narrow class groups of some ℤₚ-extensions over the rationals