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- W3089147306 abstract "Abstract Let K be a number field and S a set of primes of K . We write <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msub> <m:mi>K</m:mi> <m:mi>S</m:mi> </m:msub> <m:mo>/</m:mo> <m:mi>K</m:mi> </m:mrow> </m:math> {K_{S}/K} for the maximal extension of K unramified outside S and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msub> <m:mi>G</m:mi> <m:mrow> <m:mi>K</m:mi> <m:mo>,</m:mo> <m:mi>S</m:mi> </m:mrow> </m:msub> </m:math> {G_{K,S}} for its Galois group. In this paper, we prove the following generalization of the Neukirch–Uchida theorem under some assumptions: “For <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mn>2</m:mn> </m:mrow> </m:mrow> </m:math> {i=1,2} , let <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msub> <m:mi>K</m:mi> <m:mi>i</m:mi> </m:msub> </m:math> {K_{i}} be a number field and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msub> <m:mi>S</m:mi> <m:mi>i</m:mi> </m:msub> </m:math> {S_{i}} a set of primes of <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msub> <m:mi>K</m:mi> <m:mi>i</m:mi> </m:msub> </m:math> {K_{i}} . If <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msub> <m:mi>G</m:mi> <m:mrow> <m:msub> <m:mi>K</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mi>S</m:mi> <m:mn>1</m:mn> </m:msub> </m:mrow> </m:msub> </m:math> {G_{K_{1},S_{1}}} and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msub> <m:mi>G</m:mi> <m:mrow> <m:msub> <m:mi>K</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mi>S</m:mi> <m:mn>2</m:mn> </m:msub> </m:mrow> </m:msub> </m:math> {G_{K_{2},S_{2}}} are isomorphic, then <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msub> <m:mi>K</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> {K_{1}} and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msub> <m:mi>K</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> {K_{2}} are isomorphic.” Here the main assumption is that the Dirichlet density of <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msub> <m:mi>S</m:mi> <m:mi>i</m:mi> </m:msub> </m:math> {S_{i}} is not zero for at least one i . A key step of the proof is to recover group-theoretically the l -adic cyclotomic character of an open subgroup of <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msub> <m:mi>G</m:mi> <m:mrow> <m:mi>K</m:mi> <m:mo>,</m:mo> <m:mi>S</m:mi> </m:mrow> </m:msub> </m:math> {G_{K,S}} for some prime number l ." @default.
- W3089147306 created "2020-10-01" @default.
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- W3089147306 date "2022-02-15" @default.
- W3089147306 modified "2023-10-03" @default.
- W3089147306 title "The Neukirch–Uchida theorem with restricted ramification" @default.
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- W3089147306 doi "https://doi.org/10.1515/crelle-2021-0090" @default.
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