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Galois representations, embedding problems and modular forms.

Teresa Crespo (1997)

Collectanea Mathematica

To an odd irreducible 2-dimensional complex linear representation of the absolute Galois group of the field Q of rational numbers, a modular form of weight 1 is associated (modulo Artin's conjecture on the L-series of the representation in the icosahedral case). In addition, linear liftings of 2-dimensional projective Galois representations are related to solutions of certain Galois embedding problems. In this paper we present some recent results on the existence of liftings of projective representations...

Galois towers over non-prime finite fields

Alp Bassa, Peter Beelen, Arnaldo Garcia, Henning Stichtenoth (2014)

Acta Arithmetica

We construct Galois towers with good asymptotic properties over any non-prime finite field ℓ ; that is, we construct sequences of function fields = (N₁ ⊂ N₂ ⊂ ⋯) over ℓ of increasing genus, such that all the extensions N i / N 1 are Galois extensions and the number of rational places of these function fields grows linearly with the genus. The limits of the towers satisfy the same lower bounds as the best currently known lower bounds for the Ihara constant for non-prime finite fields. Towers with these properties...

Gaussian Integers

Yuichi Futa, Hiroyuki Okazaki, Daichi Mizushima, Yasunari Shidama (2013)

Formalized Mathematics

Gaussian integer is one of basic algebraic integers. In this article we formalize some definitions about Gaussian integers [27]. We also formalize ring (called Gaussian integer ring), Z-module and Z-algebra generated by Gaussian integer mentioned above. Moreover, we formalize some definitions about Gaussian rational numbers and Gaussian rational number field. Then we prove that the Gaussian rational number field and a quotient field of the Gaussian integer ring are isomorphic.

Généralisation d’un théorème d’Iwasawa

Jean-François Jaulent (2005)

Journal de Théorie des Nombres de Bordeaux

Nous généralisons à certains quotients finis d’un Λ -module noethérien non nécessairement de torsion le classique théorème d’Iwasawa sur l’expression asymptotique du ℓ -nombre de classes dans les ℤ ℓ -extensions. Puis nous illustrons les résultats obtenus en déterminant explicitement les caractères invariants attachés aux ℓ -groupes de S -classes T -infinitésimales dans une tour cyclotomique à partir de quelques paramètres référents et de données galoisiennes simples des extensions considérées. Un outil...

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