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Polynomial cycles in certain local domains

T. Pezda (1994)

Acta Arithmetica

1. Let R be a domain and f ∈ R[X] a polynomial. A k-tuple x ₀ , x ₁ , . . . , x k - 1 of distinct elements of R is called a cycle of f if f ( x i ) = x i + 1 for i=0,1,...,k-2 and f ( x k - 1 ) = x ₀ . The number k is called the length of the cycle. A tuple is a cycle in R if it is a cycle for some f ∈ R[X]. It has been shown in [1] that if R is the ring of all algebraic integers in a finite extension K of the rationals, then the possible lengths of cycles of R-polynomials are bounded by the number 7 7 · 2 N , depending only on the degree N of K. In this note we consider...

Prime ideal factorization in a number field via Newton polygons

Lhoussain El Fadil (2021)

Czechoslovak Mathematical Journal

Let K be a number field defined by an irreducible polynomial F ( X ) ∈ ℤ [ X ] and ℤ K its ring of integers. For every prime integer p , we give sufficient and necessary conditions on F ( X ) that guarantee the existence of exactly r prime ideals of ℤ K lying above p , where F ¯ ( X ) factors into powers of r monic irreducible polynomials in 𝔽 p [ X ] . The given result presents a weaker condition than that given by S. K. Khanduja and M. Kumar (2010), which guarantees the existence of exactly r prime ideals of ℤ K lying above p . We further specify...

Prime to p fundamental groups and tame Galois actions

Mark Kisin (2000)

Annales de l'institut Fourier

We show that for a local, discretely valued field F , with residue characteristic p , and a variety 𝒰 over F , the map ρ : Gal ( F sep / F ) → Out ( π 1 , geom ( p ' ) ( 𝒰 ) ) to the outer automorphisms of the prime to p geometric étale fundamental group of 𝒰 maps the wild inertia onto a finite image. We show that under favourable conditions ρ depends only on the reduction of 𝒰 modulo a power of the maximal ideal of F . The proofs make use of the theory of logarithmic schemes.

Propriétés du groupe tannakien des structures de Hodge p -adiques et torseur entre cohomologies cristalline et étale

Jean-Pierre Wintenberger (1997)

Annales de l'institut Fourier

On donne des propriétés de la catégorie tannakienne des modules de Dieudonné filtrés sur un corps p -adique (ces modules de Dieudonné jouent en p -adique un rôle analogue aux structures de Hodge complexes). On prouve l’existence d’un foncteur fibre sur Q p et la simple connexité du groupe associé. Ceci permet de montrer, sous la conjecture de Fontaine : “faiblement admissible entraîne admissible”, une conjecture de Rapoport et Zink décrivant le torseur entre cohomologie cristalline et étale, et de prouver...

Currently displaying 41 – 60 of 65