Parity of ranks of elliptic and hyperelliptic curves

研究领域：数论多项式方程的有理解的研究是代数数论的核心课题。阿贝尔变量（AVs）是那些解形成阿贝尔群的方程：有一种方法可以从已知解中创建新解。最经典和最简单的例子是椭圆曲线，如y^2+y=x^3-x，其中平凡解x=y=0可以生成无限个新解。尽管Birch-Swinnerton-Dyer （BSD）猜想为提取关键的算术数据（“秩”）提供了一个未经证实的方法，但关于椭圆曲线和av的核心问题仍然没有得到解决。这个猜想被认为是现代数学中未解决的核心问题之一，是七个克莱千年问题之一（现在是六个，在庞加莱猜想被佩雷尔曼解决之后）。本课题是对椭圆曲线的秩宇称和超椭圆曲线的高维AVs -雅可比矩阵的新研究。椭圆曲线理论比较发达，可以作为研究高维av的试验场。在过去几年中，超椭圆曲线取得了许多突破，这将为本项目提供非常有用的工具：这些工具包括处理根数（M. Bisatt）， Tamagawa数（a . Betts），微分（S. Kunzweiler）和局部伽罗瓦表示（T. Dokchitser， V. Dokchitser， C. Maistret, a . Morgan）的显式方法。此外，椭圆曲线和超椭圆曲线都具有适合进行数值实验的特点，可以为理论工作提供支持和指导。该项目的总体目标是在显式数论的背景下开发与BSD猜想相关的高属曲线及其不变量的新技术。具体目标是：(i)研究曲线族的秩宇称分布，（ii）开发利用不同属曲线之间的映射来控制重要算术数据（如秩和Selmer群）的方法。

Research Area: Number TheoryThe study of rational solutions to polynomial equations is a subject that is at the heart of algebraic number theory. Abelian varieties (AVs) are those equations for which the solutions form an abelian group: there is a method for creating new solutions from known ones. The classical and simplest case is that of elliptic curves, such as y^2+y=x^3-x, where the trivial solution x=y=0 lets one generate an infinite number of new solutions. The central questions about elliptic curves and AVs remain unresolved, although the Birch-Swinnerton-Dyer (BSD) conjecture provides an unproven recipe for extracting the critical arithmetic data (the "rank"). This conjecture is regarded as one of the central unsolved problems in modern mathematics, and is one of the seven Clay Millenium Problems (now six, after the Poincare conjecture was solved by Perelman).The project is a new investigation into the parity of ranks of elliptic curves and higher dimensional AVs - Jacobians of hyperelliptic curves. The theory of elliptic curves is relatively well-developed and can be used as a testing ground when studying higher dimensional AVs. Hyperelliptic curves have seen a number of breakthroughs in the last few years, which will provide very useful tools for this project: these include explicit methods for working with root numbers (M. Bisatt), Tamagawa numbers (A. Betts), differentials (S. Kunzweiler) and local Galois representations (T. Dokchitser, V. Dokchitser, C. Maistret, A. Morgan). Furthermore, both elliptic and hyperelliptic curves share the feature that they are suitable for doing numerical experiments, which can often both support and help to guide the theoretical work.The general aim of the project is to develop new techniques for working with higher genus curves and their invariants related to the BSD conjecture in the context of explicit number theory. Specific goals are to: (i) investigate the distribution of the parity of ranks for families of curves, and (ii) develop methods that exploit maps between curves of different genus to control important arithmetic data, such as ranks and Selmer groups.