Period Domains, Motives, and Ramification Theory in Arithmetic Geometry

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（或有理数）解，试图通过它们的几何结构更好地理解数字。算术几何中的著名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”）。pi和其他人开发的新技术——包括任意henselian估值环的上层分支群，动机高度，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性问题上取得重大进展。该项目还支持pi为在类似领域工作的研究人员撰写书籍，并为研究生提供研究培训机会。本课题拟加强算术几何中与这些学科相联系的格式和矢束的分支理论、各种扩展周期域和数域动机的研究，并研究相关问题(动机高度、Hodge结构的变高、Hodge理论、Nevanlinna理论、zeta函数、Tamagawa数猜想、Iwasawa理论、Sharifi猜想、物理学中出现的Beilinson调节器和周期积分的渐近行为等)。PI定义了动机的高度，并公式化了关于代数变量点高度的Manin猜想和Vojta猜想的动机版本。他打算在这些动机版本上得到不平凡的结果。PI开创了Hodge理论Nevanliina理论。他打算在这一观点的基础上对霍奇理论进行新的发展。PI将与联合首席研究员T. Fukaya合作，研究Sharifi猜想和研究非交换环的算法。该奖项反映了美国国家科学基金会的法定使命，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。

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.