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.

研究人员将研究两个与算术测量学相关的项目。第一个项目是研究各种代数家族中理性点的存在，并构建理性数字领域的Diophantine子集的库，在这种情况下，Adiophantine设置为一组合理参数，该族人在一个家庭中相应的品种具有理性点。特别是，调查员将探索其纤维是聊天室表面或更复杂的圆锥捆的家庭，因为Brauer-Manin障碍物会产生有趣的肉食。 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代数数字，特别是为了证明在自然发生的情况下，有一个可数字结合以外的封闭点，如各种结构所需的所需。示例包括在非固定品种中的融合曲线，模量空间的ofabelian品种对雅各布式是雅各比式的种类，在纤维跳跃的品种的基础上，纤维跳跃的数量，亚属理论的次数均可依次依赖于evaripessemoreties ofvaristies ofvaristies ceptriesties ceptripties。 Andalgebraic几何形状：像代数几何形状一样，它研究了解决方案多变量多项式方程式，但是在溶液的坐标（如-37）或合理数字（类似于3/5）或某些其他传统数字的元素中，解决方案的坐标（如-37）或合理数字的坐标（如-37）或可能的元素是相同数字或其他系统不同的元素，它确实如此。自古希腊人时代以来，研究了这种问题，因为它们的内在兴趣。 The Investigator的研究不是关注这些应用程序，而是关注这些应用程序的基础和周围的问题，例如，是否有可能编写计算机程序todecide的疑问，是否有合理的多种多样性方程是否具有合理数量。这项赠款涵盖的方程家族中的授予模式涵盖了，以期推断出适度的答案，同时也证明了解决方案的存在，在更大的数字系统中无限地限制了许多限制。