Rational points on varieties in families, and countable unions of varieties over countable fields

科内品种的有理点以及可数域内品种的可数并集

基本信息

批准号：
0841321
负责人：
Bjorn Poonen
金额：
$ 29.03万
依托单位：
Massachusetts Institute of Technology
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2008
资助国家：
美国
起止时间：
2008-07-01 至 2012-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=0841321&HistoricalAwards=false
关键词：
Rational points varieties families countable

项目摘要

The investigator will work on two projects connected with arithmeticgeometry. The first project is to study the existence of rationalpoints in algebraic families of varieties, and to build a library ofdiophantine subsets of the field of rational numbers, where adiophantine set in this context means the set of rational parametervalues for which the corresponding variety in a family has a rationalpoint. In particular, the investigator will explore families whosefibers are Chatelet surfaces or more complicated conic bundles, forwhich the Brauer-Manin obstruction produces interesting diophantinesets. A long-term goal of the first project is to construct a modelof the integers using diophantine sets, because this would disprove aconjecture of Mazur regarding topology of rational points andsimultaneously prove the undecidability of the problem of decidingwhether a multivariable polynomial equation has a rational solution.The second project is to study countable unions of subvarieties over acountable algebraically closed field, such as the field of algebraicnumbers, and in particular to prove that in naturally occurringsituations, there exists a closed point outside the countable union,as required for various constructions. Examples include the union ofrational curves in a non-uniruled variety, the moduli space locus ofabelian varieties isogenous to a Jacobian, the locus in the base of afamily of varieties where the Picard number of the fiber jumps, andunions of subvarieties arising from iteration of endomorphisms ofvarieties.Arithmetic geometry lies at the intersection of number theory andalgebraic geometry: like algebraic geometry, it studies the solutionsto multivariable polynomial equations, but it does so under thenumber-theoretic restriction that the coordinates of the solutions beintegers (whole numbers like -37) or rational numbers (fractions like-3/5) or perhaps elements of some other number system different fromthe traditional systems of real numbers or complex numbers. Suchquestions were studied for their intrinsic interest since the time ofthe ancient Greeks, and in the 20th century they found unforeseenapplications to cryptography and error-correcting codes. Theinvestigator's research focuses not on these applications, but on thefundamental questions underlying and surrounding them, such as thequestion of whether it is possible to write a computer program todecide whether an arbitrary multivariable polynomial equation has asolution in rational numbers. The research covered by this grant willstudy patterns in families of equations in the hope of deducing anegative answer, while also proving the existence of solutionssatisfying infinitely many constraints in a larger number system.

研究人员将致力于两个与算术几何相关的项目。第一个项目是研究簇代数族中有理点的存在性，并建立一个有理数域的丢番图子集库，这里的丢番图集是指族中相应簇的有理参数值的集合有一个合理点。特别是，研究人员将探索纤维为夏特莱曲面或更复杂的圆锥束的族，布劳尔-马宁障碍会产生有趣的丢番图集。第一个项目的长期目标是使用丢番图集构造一个整数模型，因为这将反驳马祖尔关于有理点拓扑的猜想，同时证明多元多项式方程是否有有理解的问题的不可判定性。第二个项目是研究可数代数闭域（例如代数数域）上的子族的可数并集，特别是证明在自然发生的情况下，根据各种构造的需要，在可数并集之外存在一个闭点。例子包括非单规簇中的有理曲线的并集、与雅可比行列式同源的阿贝尔簇的模空间轨迹、簇族基部中纤维皮卡德数跳跃的轨迹，以及由自同态迭代产生的子簇的并集算术几何位于数论和代数几何的交叉点：像代数几何一样，它研究多元多项式的解方程，但它是在数论限制下这样做的，即解的坐标是整数（像-37这样的整数）或有理数（像-3/5这样的分数）或者可能是不同于传统实数系统的其他数字系统的元素数字或复数。自古希腊时代以来，人们就因其内在兴趣而对此类问题进行了研究，并且在 20 世纪，它们在密码学和纠错码中发现了意想不到的应用。研究者的研究重点不是这些应用，而是它们背后和周围的基本问题，例如是否可以编写计算机程序来确定任意多元多项式方程是否有有理数解。这项资助所涵盖的研究将研究方程组中的模式，以期得出否定答案，同时证明在更大的数字系统中满足无限多个约束的解的存在性。