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Random Thue and Fermat equations

Rainer Dietmann, Oscar Marmon (2015)

Acta Arithmetica

We consider Thue equations of the form a x k + b y k = 1 , and assuming the truth of the abc-conjecture, we show that almost all locally soluble Thue equations of degree at least three violate the Hasse principle. A similar conclusion holds true for Fermat equations a x k + b y k + c z k = 0 of degree at least six.

Rang de courbes elliptiques avec groupe de torsion non trivial

Odile Lecacheux (2003)

Journal de théorie des nombres de Bordeaux

On construit des courbes elliptiques sur ℚ ( T ) de rang au moins 3, avec un sous-groupe de torsion non trivial. Par spécialisation, des courbes elliptiques de rang 5 et 6 sur ℚ sont obtenues.

Rational fixed points for linear group actions

Pietro Corvaja (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We prove a version of the Hilbert Irreducibility Theorem for linear algebraic groups. Given a connected linear algebraic group G , an affine variety V and a finite map π : V → G , all defined over a finitely generated field κ of characteristic zero, Theorem 1.6 provides the natural necessary and sufficient condition under which the set π ( V ( κ ) ) contains a Zariski dense sub-semigroup Γ ⊂ G ( κ ) ; namely, there must exist an unramified covering p : G ˜ → G and a map θ : G ˜ → V such that π ∘ θ = p . In the case κ = ℚ , G = 𝔾 a is the additive group, we reobtain the...

Rational periodic points for quadratic maps

Jung Kyu Canci (2010)

Annales de l’institut Fourier

Let K be a number field. Let S be a finite set of places of K containing all the archimedean ones. Let R S be the ring of S -integers of K . In the present paper we consider endomorphisms of ℙ 1 of degree 2 , defined over K , with good reduction outside S . We prove that there exist only finitely many such endomorphisms, up to conjugation by PGL 2 ( R S ) , admitting a periodic point in ℙ 1 ( K ) of order > 3 . Also, all but finitely many classes with a periodic point in ℙ 1 ( K ) of order 3 are parametrized by an irreducible curve.

Rational points and Coxeter group actions on the cohomology of toric varieties

Gustav I. Lehrer (2008)

Annales de l’institut Fourier

We derive a simple formula for the action of a finite crystallographic Coxeter group on the cohomology of its associated complex toric variety, using the method of counting rational points over finite fields, and the Hodge structure of the cohomology. Various applications are given, including the determination of the graded multiplicity of the reflection representation.

Rational points on a subanalytic surface

Jonathan Pila (2005)

Annales de l’institut Fourier

Let X ⊂ ℝ n be a compact subanalytic surface. This paper shows that, in a suitable sense, there are very few rational points of X that do not lie on some connected semialgebraic curve contained in X .

Currently displaying 1081 – 1100 of 1550