Algebraic and Analytic Methods in Number Theory.

算术几何中的一个基本问题是描述定义在数域K（Q的有限扩张）上的代数簇（如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亏格的性质进行分类。对于亏格1的曲线有大量的理论和方法，最近已经扩展到更高亏格和相关变种的曲线，称为雅可比变种。这方面的工作大部分在高亏格曲线和其他代数簇（如下降技术，调查的Shafarevich-Tate群和Brauer-Manin障碍）主要使用代数技术。解析数论利用技术分析，如圆的方法，并强调问题的分布数论对象，一个例子是数量N_V（B）的增长率，表示位于V上的K-有理数点的数量，投影高度由B限定。有一些假说，如马宁猜想，试图描述这种增长率。分析方法也被用来调查问题的平均职级和类数的问题。在分析数论中有大量的研究都是利用亏格1的曲线。本项目的主要目的是研究以下两个方面的问题：Abel簇（如高亏格曲线的Jacobian）的最新显式技术和数域、有理点分布和平均秩的分析问题。这些领域之间的相互作用将产生巨大的效益和创新。对于高亏格曲线上的覆盖技术，我们将研究覆盖曲线的Jacobian的平均秩，我们将研究Abel簇上挠的增长率，我们将把已有的亏格1曲线的应用推广到高亏格，这些应用提供了关于类数的分析结果;我们还将研究代数簇上有理点的分布，这需要熟悉分析技术和基础代数几何的广泛知识。目标包括一系列算术几何中的基本问题：有些强调代数技巧，有些强调分析技巧，有些则混合使用这些技巧。具体来说，在第一个18个月，我们将开发新的显式模型的齐次空间和中间对象，扭曲的库默品种，涉及到乘以m的雅可比，和其他isogenies，证明新的情况下，阿廷猜想，初步实验和特殊情况下的新的合理扭转，实验证据的特殊情况下，马宁猜想。在剩下的24个月里，我们将研究齐性空间的中间对象上的Brauer-Manin阻塞和Shafarevich-Tate群的成员之间的显式路径，可被p > 7整除的类群的结果，使用高阶亏格曲线的Jacobian的挠，具有大有理挠的Abel簇的新序列的推导，以及Manin猜想的进一步新情况的证明。这也是该项目的博士后研究助理和研究生，无论是由该项目资助的目的，应该获得在算术几何代数和分析技术的全面知识。除了新的理论，该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提供试验场。它还打算编写与上述目标相关的岩浆程序，并将其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.