Verification of strong BSD for elliptic curves and abelian surfaces over totally real number fields

全实数域上椭圆曲线和阿贝尔曲面的强 BSD 验证

• 批准号: 441241343
• 负责人: Professor Dr. Michael Stoll
• 金额: --
• 依托单位: Lehrstuhl Mathematik II - Computeralgebra, Algorithmische Arithmetische Geometrie
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/441241343?language=en
• 关键词: Verification; strong; BSD; elliptic; curves

The focus of this project is the conjecture of Birch and Swinnerton-Dyer (BSD for short) for elliptic curves over totally real number fields. An elliptic curve is an algebraic curve that carries a group structure. This means that we can add two points on the curve to get another point on the curve, and this addition has similar properties as the standard addition. Elliptic curves are important in various contexts within mathematics, for example in the proof of Fermat's Last Theorem or in cryptography.A totally real number field is an extension of the field Q of rational numbers that is generated by a root of a polynomial with rational coefficients all of whose roots are real numbers.Let E be an elliptic curve over a number field F.Using the numbers of points on E modulo each prime ideal of F, one can construct a certain function, the L-function of E. The BSD conjecture for E proposes a surprising connection between the analytic behavior of the L-function of E and certain "global" invariants of E. These invariants include properties of the group of F-rational points on E on the one hand and the number of elements of the mysterious Shafarevich-Tate group Sha(E/F) of E on the other hand. Since all other quantities that occur in the conjecture can be computed for a given E, the conjecture can be expressed as "Sha(E/F) is finite and has the expected number of elements".Birch and Swinnerton-Dyer originally formulated their conjecture for elliptic curves over Q. To prove this version is one of the seven "Millennium Problems" of the Clay Foundation.For general elliptic curves, the conjecture is wide open. It is not even known that Sha(E/F) is always finite. For so-called "modular" elliptic curves with additional properties, some parts of the conjecture are known, however, in particular the finiteness of Sha(E/F). Every elliptic curve defined over Q is modular, and so it was possible to verify the BSD conjecture for many individual elliptic curves over Q. In the predecessor of this project, we extended this to some modular abelian surfaces over Q, which are two-dimensional analogues of elliptic curves.The goal of this new project is to obtain the complete verification of the BSD conjecture also for many modular elliptic curves (and, if possible, also abelian surfaces) over totally real number fields F other than Q.The algorithms that we will develop and the data on Sha(E/F) that will result will also be useful outside the framework of this project.

本课题的重点是Birch和Swinnerton-Dyer（简称BSD）关于全实数域上椭圆曲线的猜想。椭圆曲线是一种带有群结构的代数曲线。这意味着我们可以将曲线上的两个点相加得到曲线上的另一个点，这个加法和标准加法有相似的性质。椭圆曲线在数学的各种背景下都很重要，例如在费马大定理的证明或密码学中。全实数域是有理数域Q的扩展，它是由有理数系数多项式的根生成的，其根都是实数。设E是数域F上的一条椭圆曲线。利用E上对F的每个素数理想取模的点的个数，可以构造一个函数，E的BSD猜想提出了E的l -函数的解析行为与E的某些“整体”不变量之间的惊人联系。这些不变量一方面包括E上的F有理点群的性质，另一方面包括E的神秘的shafarevic - tate群Sha（E/F）的元素数目。由于猜想中出现的所有其他数量都可以对给定的E进行计算，因此猜想可以表示为“Sha（E/F）是有限的，并且具有期望的元素数量”。Birch和Swinnerton-Dyer最初提出了关于q上椭圆曲线的猜想。为了证明这个猜想是Clay基金会的七个“千年问题”之一。对于一般椭圆曲线，这个猜想是完全开放的。我们甚至不知道Sha（E/F）总是有限的。然而，对于具有附加性质的所谓“模”椭圆曲线，猜想的某些部分是已知的，特别是Sha（E/F）的有限性。每一条定义在Q上的椭圆曲线都是模的，因此有可能对Q上的许多单独的椭圆曲线验证BSD猜想。在本项目的前作中，我们将其推广到Q上的一些模阿贝尔曲面，它们是椭圆曲线的二维类似物。这个新项目的目标是在除q以外的全实数域F上对许多模椭圆曲线（如果可能的话，也包括阿贝尔曲面）获得BSD猜想的完整验证。我们将开发的算法和Sha（E/F）上的数据也将在这个项目的框架之外有用。