**abstract:**
Given a self map $f$ of a projective variety $X$ over a number field, one typically expects the iterated preimages of a subvariety to become more complicated. An arithmetic incarnation of this, asked by Matsuzawa, Meng, Shibata, and Zhang is if the rational points of these preimages stabilize. We answer their question for etale maps and relate their problem to the following cancellation statement: when does there exist $N$ such that for all $n\geq N$ and all rational points $x$ and $y$, if $f^{n}(x)=f^{n}(y)$, then $f^{N}(x)=f^{N}(y)$? We show such cancellation phenomena hold for smooth projective curves. This is joint work with Jason Bell and Yohsuke Matsuzawa.

Mon 20 Jun, 16:20 - 17:20, Aula Dini

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