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Chebyshev's method for number fields

José Felipe Voloch (2000)

Journal de théorie des nombres de Bordeaux

We give an elementary proof of an explicit estimate for the number of primes splitting completely in an extension of the rationals. The proof uses binomial coefficents and extends Chebyshev's classical approach.

Circular units of real abelian fields with four ramified primes

Vladimír Sedláček (2017)

Archivum Mathematicum

In this paper we study the groups of circular numbers and circular units in Sinnott’s sense in real abelian fields with exactly four ramified primes under certain conditions. More specifically, we construct ℤ -bases for them in five special infinite families of cases. We also derive some results about the corresponding module of relations (in one family of cases, we show that the module of Ennola relations is cyclic). The paper is based upon the thesis [6], which builds upon the results of the paper...

Class groups of abelian fields, and the main conjecture

Cornelius Greither (1992)

Annales de l'institut Fourier

This first part of this paper gives a proof of the main conjecture of Iwasawa theory for abelian base fields, including the case p = 2 , by Kolyvagin’s method of Euler systems. On the way, one obtains a general result on local units modulo circular units. This is then used to deduce theorems on the order of χ -parts of p -class groups of abelian number fields: first for relative class groups of real fields (again including the case p = 2 ). As a consequence, a generalization of the Gras conjecture is stated...

Class groups of large ranks in biquadratic fields

Mahesh Kumar Ram (2024)

Czechoslovak Mathematical Journal

For any integer n > 1 , we provide a parametric family of biquadratic fields with class groups having n -rank at least 2. Moreover, in some cases, the n -rank is bigger than 4.

Class invariants and cyclotomic unit groups from special values of modular units

Amanda Folsom (2008)

Journal de Théorie des Nombres de Bordeaux

In this article we obtain class invariants and cyclotomic unit groups by considering specializations of modular units. We construct these modular units from functional solutions to higher order q -recurrence equations given by Selberg in his work generalizing the Rogers-Ramanujan identities. As a corollary, we provide a new proof of a result of Zagier and Gupta, originally considered by Gauss, regarding the Gauss periods. These results comprise part of the author’s 2006 Ph.D. thesis [6] in which...

Class invariants by Shimura's reciprocity law

Alice Gee (1999)

Journal de théorie des nombres de Bordeaux

We apply the Shimura reciprocity law to determine when values of modular functions of higher level can be used to generate the Hilbert class field of an imaginary quadratic field. In addition, we show how to find the corresponding polynomial in these cases. This yields a proof for conjectural formulas of Morain and Zagier concerning such polynomials.

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