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Caractérisation d'un ensemble généralisant l'ensemble des nombres de Pisot

Toufik Zaïmi (1998)

Acta Arithmetica

1. Introduction. Soient K un corps de nombres et θ un entier algébrique de module > 1 et de polynôme minimal Irr(θ,K,z) sur K. Alors θ est dit K-nombre de Pisot si pour tout plongement σ de K dans ℂ le polynôme σIrr(θ,K,z) possède une unique racine de module > 1 et aucune racine de module 1. Ces nombres ont été définis par A. M. Bergé et J. Martinet [2]. Comme dans [2], on représente un K-nombre de Pisot θ dans l’algèbre A = ℝ r ₁ × ℂ r ₂ , où (r₁,r₂) désigne la signature du corps K, par la suite ( θ σ ) σ de ses...

Catalan’s conjecture

Yuri F. Bilu (2002/2003)

Séminaire Bourbaki

The subject of the talk is the recent work of Mihăilescu, who proved that the equation x p - y q = 1 has no solutions in non-zero integers x , y and odd primes p , q . Together with the results of Lebesgue (1850) and Ko Chao (1865) this implies the celebratedconjecture of Catalan (1843): the only solution to x u - y v = 1 in integers x , y > 0 and u , v > 1 is 3 2 - 2 3 = 1 . Before the work of Mihăilescu the most definitive result on Catalan’s problem was due to Tijdeman (1976), who proved that the solutions of Catalan’s equation are bounded by an absolute...

Certain L-functions at s = 1/2

Shin-ichiro Mizumoto (1999)

Acta Arithmetica

Introduction. The vanishing orders of L-functions at the centers of their functional equations are interesting objects to study as one sees, for example, from the Birch-Swinnerton-Dyer conjecture on the Hasse-Weil L-functions associated with elliptic curves over number fields.    In this paper we study the central zeros of the following types of L-functions:    (i) the derivatives of the Mellin transforms of Hecke eigenforms for SL₂(ℤ),    (ii) the Rankin-Selberg...

Chebotarev sets

Hershy Kisilevsky, Michael O. Rubinstein (2015)

Acta Arithmetica

We consider the problem of determining whether a set of primes, or, more generally, prime ideals in a number field, can be realized as a finite union of residue classes, or of Frobenius conjugacy classes. We give necessary conditions for a set to be realized in this manner, and show that the subset of primes consisting of every other prime cannot be expressed in this way, even if we allow a finite number of exceptions.

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