English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

Given a finite endomorphism $\varphi$ of a variety $X$ defined over the fieldof fractions $K$ of a Dedekind domain, we study the extension$K(\varphi^{-\infty}(\alpha)) : = \bigcup_{n \geq 1} K(\varphi^{-n}(\alpha))$generated by the preimages of $\alpha$ under all iterates of $\varphi$. Inparticular when $\varphi$ is post-critically finite, i.e., there exists anon-empty, Zariski-open $W \subseteq X$ such that $\varphi^{-1}(W) \subseteq W$and $\varphi : W \to X$ is \'etale, we prove that$K(\varphi^{-\infty}(\alpha))$ is ramified over only finitely many primes of$K$. This provides a large supply of infinite extensions with restrictedramification, and generalizes results of Aitken-Hajir-Maire in the case $X =\mathbb{A}^1$ and Cullinan-Hajir, Jones-Manes in the case $X = \mathbb{P}^1$.Moreover, we conjecture that this finite ramification condition characterizespost-critically finite morphisms, and we give an entirely new result showingthis for $X = \mathbb{P}^1$. The proof relies on Faltings' theorem and a localargument.

Finite ramification for preimage fields of postcritically finite morphisms