On the class numbers of cyclotomic fields.
On the class numbers of real cyclotomic fields of conductor pq
The class numbers h⁺ of the real cyclotomic fields are very hard to compute. Methods based on discriminant bounds become useless as the conductor of the field grows, and methods employing Leopoldt's decomposition of the class number become hard to use when the field extension is not cyclic of prime power. This is why other methods have been developed, which approach the problem from different angles. In this paper we extend one of these methods that was designed for real cyclotomic fields of prime...
On the classgroups of imaginary abelian fields
Let be an odd prime, an odd, -adic Dirichlet character and the cyclic imaginary extension of associated to . We define a “-part” of the Sylow -subgroup of the class group of and prove a result relating its -divisibility to that of the generalized Bernoulli number . This uses the results of Mazur and Wiles in Iwasawa theory over . The more difficult case, in which divides the order of is our chief concern. In this case the result is new and confirms an earlier conjecture of G....
On the classification of Hilbert Modular threefolds.
On the common index divisors of a dihedral field of prime degree.
On the computation of quadratic -class groups
We describe an algorithm due to Gauss, Shanks and Lagarias that, given a non-square integer mod and the factorization of , computes the structure of the -Sylow subgroup of the class group of the quadratic order of discriminant in random polynomial time in .
On the diophantine equation
On the exponent of the ideal class groups of imaginary extensions of
On the Factorization of the Relative Class Number in Terms of Frobenius Divisions.
On the generalized Bernoulli numbers that belong to unequal characters.
The study of class number invariants of absolute abelian fields, the investigation of congruences for special values of L-functions, Fourier coefficients of half-integral weight modular forms, Rubin's congruences involving the special values of L-functions of elliptic curves with complex multiplication, and many other problems require congruence properties of the generalized Bernoulli numbers (see [16]-[18], [12], [29], [3], etc.). The first steps in this direction can be found in the papers of...
On the genus group of algebraic number fields
On the group of circular units of any compositum of quadratic fields
On the Hilbert -class field tower of some abelian -extensions over the field of rational numbers
It is well known by results of Golod and Shafarevich that the Hilbert -class field tower of any real quadratic number field, in which the discriminant is not a sum of two squares and divisible by eight primes, is infinite. The aim of this article is to extend this result to any real abelian -extension over the field of rational numbers. So using genus theory, units of biquadratic number fields and norm residue symbol, we prove that for every real abelian -extension over in which eight primes...
On the Hilbert -class field tower of some imaginary biquadratic number fields
Let be an imaginary bicyclic biquadratic number field, where is an odd negative square-free integer and its second Hilbert -class field. Denote by the Galois group of . The purpose of this note is to investigate the Hilbert -class field tower of and then deduce the structure of .
On the ideal class groups of the maximal real subfields of number fields with all roots of unity
In this paper, for a totally real number field we show the ideal class group of is trivial. We also study the -component of the ideal class group of the cyclotomic -extension.
On the imbeddings of imaginary quadratic orders in definite quaternion orders.
On the l-divisibility of the relative class number of certain cyclic number fields
On the l-divisibility of the relative class number of certain cyclic number fields
On the nonessential discriminant divisor of an algebraic number field
On the number of rational points of Jacobians over finite fields
We prove lower and upper bounds for the class numbers of algebraic curves defined over finite fields. These bounds turn out to be better than most of the previously known bounds obtained using combinatorics. The methods used in the proof are essentially those from the explicit asymptotic theory of global fields. We thus provide a concrete application of effective results from the asymptotic theory of global fields and their zeta functions.