Algebraic and Analytic Methods in Number Theory.

算术几何的基本问题之一是描述在数域K （Q的有限扩展）上定义的代数变量（如由多项式方程描述的曲线和曲面）上的有理点。代数曲线可以根据一个叫做“属”的性质来分类。有大量的理论和方法的曲线属1，最近已扩展到曲线的更高的属和相关的变种，被称为雅可比变量。许多关于高属曲线和其他代数变种的工作（如下降技术，对shafarevic - tate群和Brauer-Manin阻塞的研究）主要使用代数技术。解析数论利用分析中的技术，如圆法，并强调数论对象的分布问题，如质数或有理点在各种V上的分布。一个例子是数量N_V(B)的增长率，表示投影高度以B为界的V上k个有理点的数量。有一些猜想，如马宁猜想，试图描述这种增长率。分析方法也被用于研究平均等级和类数问题。在解析数论中有大量的研究利用了属1的曲线。在这个项目中，我们的主要目的是研究涉及以下两个问题：最近关于阿贝尔变体的显式技术（如高属曲线的雅可比矩阵）和关于数域、有理点分布和平均秩的解析问题。这些领域之间的相互作用将产生巨大的利益和创新。对于高属曲线上的覆盖技术，我们将研究覆盖曲线雅可比矩阵的平均秩；我们将研究阿贝尔变体上的扭转的增长率；我们将采用现有的1格曲线的应用，这些应用已经提供了关于类数的解析结果，并将它们推广到更高的格；我们还将研究关于代数变量上有理点分布的猜想，这需要熟悉解析技术和广泛的代数几何基础知识。目标包括一系列算术几何的基本问题：一些强调代数技术，一些强调分析技术，还有一些使用这些技术的混合。具体来说，在前18个月，我们将为齐次空间和中间对象开发新的显式模型，扭曲的Kummer变种，与雅可比矩阵上的m相乘和其他等同性有关，Artin猜想的新情况的证明，新理性扭转的初始实验和特殊情况，以及Manin猜想的特殊情况的实验证据。在剩下的24个月里，我们将研究齐次空间中间对象上的Brauer-Manin障碍与shafareovich - tate群成员之间的显式路线，关于可被p >7分的类群的结果，利用高属曲线的jacobian上的扭转，具有大合理扭转的Abelian变体的新序列的推导，以及进一步证明Manin猜想的新情况。该项目的另一个目标是，由该项目的资助的博士后研究助理和研究生应该获得算术几何中代数和分析技术的全面知识。除了新的理论，这个项目还将提供大量的实验数据，这将为其他研究人员的理论和猜想形成一个试验场。它还打算在Magma中编写与上述目标相关的程序，并将这些程序提供给其他研究人员。

One of the fundamental problems in Arithmetic Geometry is to describe the rational points on algebraic varieties (such as curves and surfaces described by polynomial equations) defined over a number field K (a finite extension of Q). Algebraic curves can be classified according to a property called genus. There is a substantial body of theory and methodology for curves of genus 1, which has recently been extended to curves of higher genus and associated varieties, known as Jacobian varieties. Much of this work on higher genus curves and other algebraic varieties (such descent techniques, investigation of the Shafarevich-Tate group and the Brauer-Manin obstruction) has used mainly algebraic techniques.Analytic Number Theory makes use of techniques from analysis, such as the circle method, and emphasises questions about distributions of number theoretic objects, such as primes or rational points on a variety V. One example is the rate of growth of the quantity N_V(B), denoting the number of K-rational points lying on V, of projective height bounded by B. There are conjectures, such as Manin's Conjecture, which attempt to describe this rate of growth. Analytic methods have also been used to investigate questions about average ranks and class number problems. There is a large body of research in Analytic Number Theory that makes use of curves of genus 1.Our main aim in this project is to investigate questions involving both of: recent explicit techniques on Abelian varieties (such as Jacobians of higher genus curves) and analytic problems on number fields, distributions of rational points, and average rank. Substantial benefits and innovations will arise from the interplay between these areas. For covering techniques on higher genus curves, we shall investigate the average rank of the Jacobians of the covering curves; we shall investigate rate of growth of torsion on Abelian varieties; we shall take existing applications of genus 1 curves which have provided analytic results about class numbers, and generalise them to higher genus; we shall also investigate conjectures on the distribution of rational points on algebraic varieties, which require both familiarity with analytic techniques and an extensive knowledge of the underlying algebraic geometry.The objectives include a range of fundamental problems in Arithmetic Geometry: some emphasising algebraic techniques, some emphasising analytic techniques, and some using a blend of these. Specifically, during the first 18 months, we shall develop new explicit models for homogeneous spaces and intermediate objects, twisted Kummer varieties, relating both to multiplication by m on Jacobians, and other isogenies, the proof of new cases of Artin's Conjecture, initial experimentations and special cases of new rational torsions, and experimental evidence on special cases of Manin's conjecture. In the remaining 24 months, we shall investigate explicit routes between the Brauer-Manin obstruction on intermediate objects of homogeneous spaces and members of the Shafarevich-Tate group, results on class groups divisible by p > 7, using torsion on Jacobians of higher genus curves, the derivation of new sequences of Abelian varieties with large rational torsion, and the proof of further new cases of Manin's Conjecture. It is also an aim of the project that the postdoctoral research associate and research student, both funded by the project, should gain a well rounded knowledge of both algebraic and analytic techniques in Arithmetic Geometry. As well as new theory, this project will also provide a substantial body of experimental data, which will form a testing ground for the theory and conjecture of other researchers. It is also intended to write programs in Magma relevant to the above objectives, and to make these available to other researchers.