Geometric constructions over finite fields, elementary equivalence of finitely generated fields, and rational points on varieties

有限域上的几何构造、有限生成域的基本等价以及簇上的有理点

基本信息

批准号：
0301280
负责人：
Bjorn Poonen
金额：
$ 37.54万
依托单位：
University of California-Berkeley
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2003
资助国家：
美国
起止时间：
2003-07-01 至 2008-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=0301280&HistoricalAwards=false
关键词：
Geometric constructions over finite fields

项目摘要

DMS-0301280Bjorn M. PoonenThe investigator proposes:1) to use sieve methods to prove the existence of various geometricobjects and maps over finite fields. For many problems, the standarddimension-based arguments work only over infinite fields; the idea isto replace them by numerical counting.2) to continue the study initiated by F. Pop on the question ofwhether elementarily equivalent finitely generated fields arenecessarily isomorphic.3) to write a graduate text highlighting the application of schemes tothe study of varieties over nonalgebraically closed fields.Originally, algebraic geometry was concerned with the real numbersolutions to systems of polynomial equations, but the 20th centurymade it clear that it was fruitful to generalize by considering fields(i.e., number systems) other than the real numbers. For example,finite fields have proved invaluable in the theory of error-correctingcodes and cryptography. Unfortunately, many geometric constructionsare known to work only over infinite fields. The investigator's firstproject is to use combinatorial techniques to adapt theseconstructions to the finite field case. The investigator's secondproject concerns the relationships between geometry over differentfields; potentially it could allow results over one field to betransferred to another field. Finally, the investigator proposes towrite a graduate text that will make newly developed techniques inthis area accessible to students.

DMS-0301280Bjorn M. Poonen 研究者提出：1）使用筛法来证明有限域上各种几何对象和映射的存在性。对于许多问题，基于标准维度的论证仅适用于无限域。这个想法是用数字计数来代替它们。2) 继续 F. Pop 发起的关于基本等价的有限生成域是否必然同构问题的研究。3) 写一篇研究生论文，重点介绍方案在非代数上的簇研究中的应用最初，代数几何关注的是多项式方程组的实数解，但 20 世纪清楚地表明，通过考虑进行推广是富有成效的除实数之外的字段（即数字系统）。例如，有限域已被证明在纠错码和密码学理论中具有无价的价值。不幸的是，许多几何构造已知只能在无限域上起作用。研究人员的第一个项目是使用组合技术使这些结构适应有限场情况。研究人员的第二个项目涉及不同领域的几何之间的关系；它可能允许将一个领域的结果转移到另一个领域。最后，研究人员建议撰写一篇研究生论文，以使学生能够接触到该领域新开发的技术。