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2023 Fiscal Year Final Research Report

Number theory of prehomogeneous vector spaces

Research Project

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Project/Area Number 20K03512
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 11010:Algebra-related
Research InstitutionTohoku University (2023)
Kyoto University (2020-2022)

Principal Investigator

Yukie Akihiko  東北大学, 理学研究科, 客員研究者 (20312548)

Project Period (FY) 2020-04-01 – 2024-03-31
Keywordsprehomogeneous / GIT / density theorem / GIT stratification
Outline of Final Research Achievements

We determined the density of unnormalized Tamgawa numbers of orthogonal groups of arbitrary number of variables over any number field. This work is a version over the field of rational numbers (which is more difficult) of Siegel's result over the ring of integers in 1944. We also determined the density of relative hR of quadratic fields and their composition with a fixed cubic field. This result concerns the 8 dimensional prehomogeneous vector space which is the most non-split version of pairs of 2x2 matricess. We also published 3 papers which determined the GIT stratification of many prehomogeneous vector spaces whose groups are products of general linear groups. By these works, we determined rational orbits of those prehomogeneous vector spaces including points where the action is not good. We also determined the inductive structures of the set of rationaal orbits.

Free Research Field

GIT and anaalytic number theory

Academic Significance and Societal Importance of the Research Achievements

最初の密度定理は Siegel の結果を 80 年ぶりに改良したものである。2番目の密度定理は相対的な状況で hR の密度を決定したあものとしては最初の結果である。密度定理はいくつか知られているが、主要項だけでは、誤差がよくないことが知られている。2番目以降の項を決定するためにはゼータ関数に関する情報がもっと必要であり、そのためには、概均質ベクトル空間の作用が悪い点も含めて、有理軌道、および帰納的構造を決定する必要がある。GIT stratification に関する結果は重要な場合について、そのような情報をもたらすものである。

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Published: 2025-01-30  

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