Uniform bounds for preperiodic points in families of twists

Levy, Alon; Manes, Michelle; Thompson, Bianca

doi:10.1090/S0002-9939-2014-12086-7

Uniform bounds for preperiodic points in families of twists
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by Alon Levy, Michelle Manes and Bianca Thompson PDF

Proc. Amer. Math. Soc. 142 (2014), 3075-3088 Request permission

Abstract:

Let $\phi$ be a morphism of $\mathbb {P}^N$ defined over a number field $K.$ We prove that there is a bound $B$ depending only on $\phi$ such that every twist of $\phi$ has no more than $B$ $K$-rational preperiodic points. (This result is analogous to a result of Silverman for abelian varieties.) For two specific families of quadratic rational maps over $\mathbb {Q}$, we find the bound $B$ explicitly.

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