Uniform bounds for preperiodic points in families of twists

@inproceedings{Levy2012UniformBF,
  title={Uniform bounds for preperiodic points in families of twists},
  author={Alon Y. Levy and Michelle Manes and Bianca Thompson},
  year={2012}
}
Letbe a morphism of PN defined over a number field K. We prove that there is a bound B depending only onsuch that every twist of � has no more than B K-rational preperiodic points. (This result is analagous to a result of Silverman for abelian varieties (10).) For two specific families of quadratic rational maps over Q, we find the bound B explicitly. 
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References

SHOWING 1-10 OF 17 REFERENCES

Rational 6-cycles under iteration of quadratic polynomials

We present a proof, which is conditional on the Birch and Swinnerton-Dyer Conjecture for a specific abelian variety, that there do not exist rational numbers x and c such that x has exact period N =…

The Space of Morphisms on Projective Space

The theory of moduli of morphisms on P^n generalizes the study of rational maps on P^1. This paper proves three results about the space of morphisms on P^n of degree d > 1, and its quotient by the…

ℚ‐rational cycles for degree‐2 rational maps having an automorphism

Let ϕ:ℙ1 → ℙ1 be a rational map of degree d = 2 defined over ℚ and assume that f−1°ϕ° f = ϕ for exactly one nontrivial f ε PGL2 (ℚ−). We describe families of such maps that have ℚ‐rational periodic…

Cycles of quadratic polynomials and rational points on a genus-$2$ curve

It has been conjectured that for N sufficiently large, there are no quadratic polynomials in Q[z] with rational periodic points of period N. Morton proved there were none with N=4, by showing that…

Questions on self maps of algebraic varieties

In this note we discuss some arithmetic and geometric questions concerning self maps of projective algebraic varieties.

Periodic Points on an Algebraic Variety

where the Li(x) are homogeneous polynomials of degree 1 with coefficients which are algebraic numbers. One of the most interesting cases arises when the following two conditions are satisfied. (1)…

Isotriviality is equivalent to potential good reduction for endomorphisms of ${\mathbb P}^N$ over function fields

Bornes pour la torsion des courbes elliptiques sur les corps de nombres

Corollaire. — Soit d un entier ≥ 1 . Il existe un nombre réel B(d) tel que pour toute courbe elliptique E , définie sur un corps de nombres K de degré d sur Q , tout point de torsion de E(K) soit…

The Arithmetic of Dynamical Systems

* Provides an entry for graduate students into an active field of research * Each chapter includes exercises, examples, and figures * Will become a standard reference for researchers in the field …

Rational periodic points of rational functions