초록

Abstract ▼

□ 연구개요
The main object of the project was the investigation of local and global behavior of Galois extensions with prescribed Galois group, with focus on the specializations of Galois covers of curves or higher-dimensional varieties. This included on the one hand (“Part 1”) the problem of realizi

□ 연구개요
The main object of the project was the investigation of local and global behavior of Galois extensions with prescribed Galois group, with focus on the specializations of Galois covers of curves or higher-dimensional varieties. This included on the one hand (“Part 1”) the problem of realizing Galois extensions of number fields with prescribed local properties, such as prescribed (inertia and) decomposition groups. On the other hand (“Part 2”), the investigation of local and global behavior of specializations of Galois covers led to arithmetic-geometric investigations and development of a new and relevant theory of “parametric dimension” and “local dimension” of a given finite group; these are invariants describing the complexity of the global, respective local Galois theory of a given group over a number field.

□ 연구 목표대비 연구결과
Regarding “Part 1”, we developed a systematic approach to obtain Galois extensions with special prescribed local behavior, in particular “locally cyclic” and “locally abelian” Galois extensions, and applied these methods to gain new insights in the Galois theory of some non-solvable groups (published in March 2021). The research in “Part 2” has resulted in several publications, of which a smaller one (dealing with explicit computation around Galois realizations of some prescribed groups) and a larger one (dealing systematically with the problem of “non-parametricity” of Galois covers of curves, from a viewpoint of arithmetic statistics) have been published (Nov.2020 and Sept.2021, respectively). The “main paper” of this part of the project, introducing for the first time in full generality the central notions of parametric and local dimension has been submitted and is under review at the present time. A further paper completing the investigations in “dimension 1” (parametricity and non-parametricity of covers of curves) is in preparation and nearing completion. Even though research on this part of the project has taken up slightly more time than initially predicted, the completeness of some of the results has exceeded original expectations.

□ 연구개발성과의 활용 계획 및 기대효과(연구개발결과의 중요성)
The results of “Part 1” have direct application to the important problem of finding unramified extensions of (low degree; e.g., quadratic) number fields with prescribed Galois group, which can be seen as a generalization of class field theory. A follow-up paper dealing directly with such unramified extensions of quadratic number field is near completion at the time this report is written. The problem of “locally cyclic” extensions will also be investigated further in an international cooperation.
The results of “Part 2”, which should be seen as the most significant outcome of this project, will lead to various further developments and cooperations with several researchers. A next stage will aim to further unify the results obtained here with related results by other researchers.

(source : 연구결과 요약문 3p)

목차 Contents

• COVER ... 1
• 연구결과 요약문 ... 3
• 목차 ... 4
• 1. 연구개발과제의 개요 ... 5
• 2. 연구개발과제의 수행 과정 및 수행 내용 ... 6
• Research methods ... 6
• Concrete research activities ... 7
• 3. 연구개발과제의 수행 결과 및 목표 달성 수준 ... 8
• 1) 정성적 연구개발성과(연구개발결과) ... 8
• 2) 세부 정량적 연구개발성과 ... 10
• 3) 목표 달성 수준 ... 11
• 4. 연구개발성과의 관련 분야에 대한 기여 정도(연구개발결과의 중요성) ... 11
• 5. 연구개발성과의 관리 및 활용 계획 ... 13
• 6. 참고문헌 ... 14
• 붙임1 세부 정량적 연구개발성과 ... 15
• 붙임2 연구책임자 대표적 연구실적 및 증빙(요약문 및 사본) ... 16
• End of Page ... 27