# 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

- Mathematics
- 2008

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

- Mathematics
- 2009

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

- Mathematics
- 2008

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

- Mathematics
- 1995

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

- Mathematics
- 2002

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

### Periodic Points on an Algebraic Variety

- Mathematics
- 1950

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

- Mathematics
- 2008

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

- Mathematics
- 1996

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

- Mathematics
- 2007

* 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
…