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A class–field theoretical calculation

Cristian D. Popescu (2006)

Journal de Théorie des Nombres de Bordeaux

In this paper, we give the complete characterization of the p –torsion subgroups of certain idèle–class groups associated to characteristic p function fields. As an application, we answer a question which arose in the context of Tan’s approach [6] to an important particular case of a generalization of a conjecture of Gross [4] on special values of L –functions.

A combinatorial interpretation of Serre's conjecture on modular Galois representations

Adriaan Herremans (2003)

Annales de l’institut Fourier

We state a conjecture concerning modular absolutely irreducible odd 2-dimensional representations of the absolute Galois group over finite fields which is purely combinatorial (without using modular forms) and proof that it is equivalent to Serre’s strong conjecture. The main idea is to replace modular forms with coefficients in a finite field of characteristic p , by their counterparts in the theory of modular symbols.

A note on free pro- p -extensions of algebraic number fields

Masakazu Yamagishi (1993)

Journal de théorie des nombres de Bordeaux

For an algebraic number field k and a prime p , define the number ρ to be the maximal number d such that there exists a Galois extension of k whose Galois group is a free pro- p -group of rank d . The Leopoldt conjecture implies 1 ρ r 2 + 1 , ( r 2 denotes the number of complex places of k ). Some examples of k and p with ρ = r 2 + 1 have been known so far. In this note, the invariant ρ is studied, and among other things some examples with ρ < r 2 + 1 are given.

À propos du théorème de Belyi

Jean-Marc Couveignes (1996)

Journal de théorie des nombres de Bordeaux

Le théorème de Belyi affirme que sur toute courbe algébrique C lisse projective et géométriquement connexe, définie sur ¯ , il existe une fonction f non ramifiée en dehors de 0 , 1 , . Nous montrons que cette fonction peut être choisie sans automorphismes, c’est-à-dire telle que pour tout automorphisme non trivial a de C , on ait f 𝔞 f . Nous en déduisons que si 𝕂 est une extension finie de , toute 𝕂 -classe d’isomorphisme de courbes algébriques lisses projectives géométriquement connexes peut être caractérisée...

Algebraic properties of a family of Jacobi polynomials

John Cullinan, Farshid Hajir, Elizabeth Sell (2009)

Journal de Théorie des Nombres de Bordeaux

The one-parameter family of polynomials J n ( x , y ) = j = 0 n y + j j x j is a subfamily of the two-parameter family of Jacobi polynomials. We prove that for each n 6 , the polynomial J n ( x , y 0 ) is irreducible over for all but finitely many y 0 . If n is odd, then with the exception of a finite set of y 0 , the Galois group of J n ( x , y 0 ) is S n ; if n is even, then the exceptional set is thin.

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