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Finite ramification for preimage fields of post-critically finite morphisms
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2017 (English)In: Mathematical Research Letters, ISSN 1073-2780, E-ISSN 1945-001X, Vol. 24, no 6, p. 1633-1647Article in journal (Refereed) Published
Abstract [en]

Given a finite endomorphism phi of a variety X defined over the field of fractions K of a Dedekind domain, we study the extension K (phi(-infinity)(alpha)) := boolean OR(n >= 1) K (phi(-n) (alpha)) generated by the preimages of alpha under all iterates of phi. In particular when phi is post-critically finite, i.e., there exists a non-empty, Zariski-open W subset of X such that phi(-1) (W) subset of W and phi : W -> X is etale, we prove that K (phi(-infinity) (alpha)) is rami fied over only finitely many primes of K. This provides a large supply of in finite extensions with restricted rami fication, and generalizes results of Aitken-Hajir-Maire [1] in the case X = A(1) and Cullinan-Hajir, Jones-Manes [7, 13] in the case X = P-1. Moreover, we conjecture that this finite rami fication condition characterizes post-critically finite morphisms, and we give an entirely new result showing this for X = P-1. The proof relies on Faltings' theorem and a local argument.

Place, publisher, year, edition, pages
International Press of Boston, Inc. , 2017. Vol. 24, no 6, p. 1633-1647
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Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-223312ISI: 000423669700003Scopus ID: 2-s2.0-85041865241OAI: oai:DiVA.org:kth-223312DiVA, id: diva2:1183222
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QC 20180216

Available from: 2018-02-16 Created: 2018-02-16 Last updated: 2018-02-16Bibliographically approved

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