Third Italian Number Theory Meeting

# Preperiodic points of rational maps defined over a global field in terms of good reduction

speaker: Laura Paladino (University of Pisa)

abstract: Let $K$ be a global field. We denote by $p$ its characteristic. Let $D$ be the degree of $K$ over the base field, that is $\mathbb{Q}$ when $p=0$ and ${\mathbb{F}}p(t)$ when $p>0$. We denote by $\phi$ an endomorphism of the projective line defined over $K$ and by $d$ its degree. A point $P\in {\mathbb{P}}1(K)$ is called \emph{periodic} for $\phi$ if there exists an integer $n>0$ such that $\phin(P)=P$. We say that $P$ is a \emph{preperiodic} point for $\phi$ if its (forward) orbit $O\phi(P)=\{\phin(P)\mid n\in{\mathbb{N}}\}$ contains a periodic point, i. e. it is finite. We show a bound for the cardinality of $O\phi(P)$ depending only on $D, p$ and the number of places of bad reduction of $\phi$. Furthermore, we show a bound for the cardinality of the set of $K$--rational preperiodic points for $\phi$, depending on $D, d, p$ and the number of places of bad reduction of $\phi$. The results are completely new in the function fields case and they improve the ones known in the number fields case. (Joint work with Jung Kyu Canci of the University of Basel)

timetable:
Mon 21 Sep, 17:00 - 17:25, Aula Contini
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