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On the classgroups of imaginary abelian fields

David Solomon (1990)

Annales de l'institut Fourier

Let p be an odd prime, χ an odd, p -adic Dirichlet character and K the cyclic imaginary extension of Q associated to χ . We define a “ χ -part” of the Sylow p -subgroup of the class group of K and prove a result relating its p -divisibility to that of the generalized Bernoulli number B 1 , χ - 1 . This uses the results of Mazur and Wiles in Iwasawa theory over Q . The more difficult case, in which p divides the order of χ is our chief concern. In this case the result is new and confirms an earlier conjecture of G....

On the computation of quadratic 2 -class groups

Wieb Bosma, Peter Stevenhagen (1996)

Journal de théorie des nombres de Bordeaux

We describe an algorithm due to Gauss, Shanks and Lagarias that, given a non-square integer D ≡ 0 , 1 mod 4 and the factorization of D , computes the structure of the 2 -Sylow subgroup of the class group of the quadratic order of discriminant D in random polynomial time in log D .

On the generalized Bernoulli numbers that belong to unequal characters.

Ilya Sh. Slavutskii (2000)

Revista Matemática Iberoamericana

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 Hilbert 2 -class field tower of some abelian 2 -extensions over the field of rational numbers

Abdelmalek Azizi, Ali Mouhib (2013)

Czechoslovak Mathematical Journal

It is well known by results of Golod and Shafarevich that the Hilbert 2 -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 2 -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 2 -extension over ℚ in which eight primes...

On the Hilbert 2 -class field tower of some imaginary biquadratic number fields

Mohamed Mahmoud Chems-Eddin, Abdelmalek Azizi, Abdelkader Zekhnini, Idriss Jerrari (2021)

Czechoslovak Mathematical Journal

Let 𝕜 = ℚ 2 , d be an imaginary bicyclic biquadratic number field, where d is an odd negative square-free integer and 𝕜 2 ( 2 ) its second Hilbert 2 -class field. Denote by G = Gal ( 𝕜 2 ( 2 ) / 𝕜 ) the Galois group of 𝕜 2 ( 2 ) / 𝕜 . The purpose of this note is to investigate the Hilbert 2 -class field tower of 𝕜 and then deduce the structure of G .

On the number of rational points of Jacobians over finite fields

Philippe Lebacque, Alexey Zykin (2015)

Acta Arithmetica

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.

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