Arakelov Theory, Motives and Special Values

阿拉克洛夫理论、动机和特殊价值观

• 批准号: 0901373
• 负责人: Henri Gillet
• 金额: $ 28万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-01 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901373&HistoricalAwards=false
• 关键词: Arakelov; Theory; Motives; Special; Values

Professor Gillet will be studying a number of problems related to Arakelov Theory, Motives, and special values of zeta and L-functions. He plans to show that there is a specialization map for motives of varieties over discretely valued fields; this would for example describe the relationship between the motive of a variety defined over the rational numbers and the associated, possibly singular, varieties over finite fields. He plans to study whether one can do arithmetic intersection theory on singular varieties, and improve our understanding of the height pairing on algebraic cycles which are homologically trivial. He also intends to study questions relating heights and similar integrals to special values of Dirichlet series. This includes understanding the relationship between heights of conics and the eigenvalues of Heun's equation, which is a classical differential equation arising from problems in physics.The overall goal of these areas of research is to improve our understanding of Diophantine equations, that is, understanding the properties of the set of integer solutions of equations with integer coefficients. Diophantine equations have applications in particular to problems in cryptography and coding theory.

吉列教授将研究与阿拉克洛夫理论、动机以及泽塔和L函数的特殊价值有关的一些问题。他计划证明在离散值域上存在变种动机的专门化映射；例如，这将描述定义在有理数上的变种的动机与有限域上相关的、可能是奇异的变种之间的关系。他计划研究人们是否可以在奇异簇上做算术交集理论，并改善我们对同调平凡的代数圈上的高度配对的理解。他还打算研究与Dirichlet级数的特殊值有关的高度和类似积分的问题。这包括了解二次曲线的高度与Heun方程的特征值之间的关系，Heun方程是一个源于物理问题的经典微分方程，这些研究领域的总体目标是提高我们对丢番图方程的理解，即理解整系数方程的整数解集的性质。丢番图方程特别适用于密码学和编码理论中的问题。