Equidistribution of Algebraic Points and Arithmetic Invariants of Modular Curves

代数点的均匀分布和模曲线的算术不变量

• 批准号: 0070686
• 负责人: Emmanuel Ullmo
• 金额: $ 10.13万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2002-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070686&HistoricalAwards=false
• 关键词: Equidistribution; Algebraic; Points; Arithmetic; Invariants

The investigator main topic is number theory. He is using Arakelov theory with a special eye on modular forms and automorphic forms. He is studying equidistribution phenomenas for sequences of algebraic points on varieties over number fields. His main result is the discreteness of the set of algebraic points of a curve of genus greater than one embedded in his Jacobian for the Neron-Tate topology. He tries to have results in the direction of the Andre-Oort conjecture concerning the possible Zariski closures of a set of points with complex multiplication in the moduli space of abelian varieties. Equidistribution of Hecke (obtained by the author and Clozel) is a step towards equidistribution of Galois orbits of points with complex multiplication. In an other direction, the investigator studies the arithmetic invariants of modular curve from the Arakelov view-point. Diophantine geometry is one of the most aesthetic subject of mathematics. Even if the questions are phrased in a very elementary way: Find all the rational or algebraic solutions of a system of polynomial equations; the solution to this type of problems requires in general the most advanced techniques of mathematics. This work explains that the set of algebraic points of a curve (which is infinite) is not very big: We can think about it as a discrete set in a natural arithmetic set.

研究者的主要课题是数论。 他使用阿拉克洛夫理论与特别着眼于模块形式和自守形式。 他正在研究equidistribution phenomenas序列的代数点品种超过一些领域。 他的主要成果是离散的一套代数点的曲线属大于一个嵌入在他的雅可比的Neron泰特拓扑。 他试图有结果的方向的安德烈-奥尔特猜想有关可能Zebriki关闭的一组点与复杂的乘法模空间的阿贝尔品种。 Hecke的等分布（由作者和Clozel获得）是向具有复数乘法的点的伽罗瓦轨道的等分布迈出的一步。 在另一个方向上，研究者从Arakelov的观点出发，研究了模曲线的算术不变量。 丢番图几何是数学中最富美感的学科之一。 即使这些问题是以非常基本的方式表达的：找到一个多项式方程组的所有有理或代数解;解决这类问题通常需要最先进的数学技术。 这项工作解释了一条曲线的代数点的集合（这是无限的）不是很大：我们可以把它看作是一个自然算术集合中的离散集合。