Period Domains, Motives, and Ramification Theory in Arithmetic Geometry

算术几何中的周期域、动机和衍生理论

• 批准号: 2001182
• 负责人: Kazuya Kato
• 金额: $ 41.4万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2001182&HistoricalAwards=false
• 关键词: Period; Domains; Motives; Ramification; Theory

This award supports the principal investigators' research in arithmetic geometry. At its heart, arithmetic geometry is concerned with integer (or rational) solutions of polynomial equations, seeking to better understand numbers through their geometric structure. Famous questions in arithmetic geometry include the celebrated Fermat's Last Theorem. In particular, the PIs will study branching behavior (called "ramification") of these geometric objects, and they will investigate parameter spaces that describe them (called "period domains"). New technology developed by the PIs and others -- including upper ramification groups of an arbitrary henselian valuation ring, heights of motives, and motive versions of the Manin conjecture and the Vojta conjecture -- should allow for major progress on open questions in these areas. The project also provides support for the PIs to write books for researchers working in similar areas and provides research training opportunities for graduate students.The principal investigator intends to strengthen his study of ramification theory of schemes and of ell-adic sheaves, various extended period domains, and motives over number fields in arithmetic geometry connecting these subjects, and to study related problems (heights of motives, heights of variation of Hodge structures, Hodge theoretic Nevanlinna theory, zeta functions, Tamagawa number conjecture, Iwasawa theory, Sharifi conjecture, asymptotic behaviors of Beilinson regulators and period integrals which appear in physics etc.). The PI defined heights of motives, and formulated motive versions of the Manin conjecture and the Vojta conjecture about heights of points of an algebraic variety. He intends to obtain non-trivial results on these motive versions. The PI started the Hodge theoretic Nevanliina theory. He plans to make new progress in Hodge theory basing on this Nevanlinna point of view. The PI will collaborate with the co-principal investigator T. Fukaya to study Sharifi conjectures and to study the arithmetic of non-commutative rings.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该奖项支持主要研究人员在算术几何方面的研究。算术几何的核心是多项式方程的整数（或有理）解，试图通过它们的几何结构更好地理解数字。算术几何中著名的问题包括著名的费马大定理。特别是，PI将研究这些几何对象的分支行为（称为“分支”），他们将研究描述它们的参数空间（称为“周期域”）。由PI和其他人开发的新技术-包括任意Henselian赋值环的上分支群，动机的高度，以及Manin猜想和Vojta猜想的动机版本-应该允许在这些领域的开放问题上取得重大进展。该项目还支持研究所为类似领域的研究人员撰写书籍，并为研究生提供研究培训机会。首席研究员打算加强对概型和ell-adic层的分歧理论、各种扩展周期域和算术几何中数域的动机的研究，并研究相关问题（动机的高度，Hodge结构的高度变化，Hodge理论的Nevanlinna理论，zeta函数，Tamagawa数猜想，Iwasawa理论，Sharifi猜想，Beilinson调节器的渐近行为和物理学中出现的周期积分等）。PI定义了动机的高度，并制定了关于代数簇的点的高度的Manin猜想和Vojta猜想的动机版本。他打算在这些动机版本上获得重要的结果。PI开创了Hodge理论的Nevanliina理论。他计划在此基础上对霍奇理论进行新的发展。主要研究者将与共同主要研究者T.该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Logarithmic abelian varieties, Part VII: Moduli

对数阿贝尔簇，第七部分：模数

• 发表时间: 2021
• 期刊: Yokohama Mathematical Journal
• 作者: Kajiwara, T.

On log motives

关于日志动机

• DOI: 10.2140/tunis.2020.2.733
• 发表时间: 2020
• 期刊: Tunisian Journal of Mathematics
• 影响因子: 0.9
• 作者: Ito, Tetsushi;Kato, Kazuya;Nakayama, Chikara;Usui, Sampei
• 通讯作者: Usui, Sampei

Logarithmic Structures of Fontaine-Illusie. II ---Logarithmic Flat Topology

Fontaine-Illusie 的对数结构。

• DOI: 10.3836/tjm/1502179316
• 发表时间: 2020
• 期刊: Tokyo Journal of Mathematics
• 影响因子: 0.6
• 作者: KATO, Kazuya

Deligne–Beilinson cohomology and log Hodge theory

Deligne–Beilinson 上同调和对数 Hodge 理论

• DOI: 10.3792/pjaa.99.006
• 发表时间: 2023
• 期刊: Mathematical Sciences
• 影响因子: 2
• 作者: Kato, Kazuya;Nakayama, Chikara;Usui, Sampei
• 通讯作者: Usui, Sampei