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

本课题的重点是关于全实数域上椭圆曲线的Birch和Swinnerton-Dyer(简称BSD)猜想。椭圆曲线是带有群结构的代数曲线。这意味着我们可以在曲线上添加两个点来获得曲线上的另一个点，并且这个加法具有与标准加法类似的性质。椭圆曲线在数学中的各种情况下都是重要的，例如在费马大定理的证明中或在密码学中。全实数域是由根都是实数的有理系数多项式的根生成的有理数域Q的扩张。设E是数域F上的椭圆曲线。利用E上的点数模F的每个素数理想，可以构造某个函数，E的L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”不变量之间惊人的联系。这些不变量一方面包括E上F-有理点群的性质，另一方面包括E的神秘Shafarevich-Tate群的元素个数(E/F)。由于猜想中出现的所有其他量都可以对给定的E进行计算，因此猜想可以表示为“Sa(E/F)是有限的，并且具有预期的元素数”。Birch和Swinnerton-Dyer最初提出了他们对Q上的椭圆曲线的猜想。以证明这个版本是粘土基金会的七个“千禧年问题”之一。对于一般的椭圆曲线，猜想是完全开放的。甚至不知道Sa(E/F)总是有限的。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Sa(E/F)的有限性。定义在Q上的每条椭圆曲线都是模的，因此可以对Q上的许多单独的椭圆曲线验证BSD猜想。在这个项目的前身中，我们将其推广到Q上的一些模交换曲面，它们是椭圆曲线的二维模拟。这个新项目的目标是获得对除Q之外的许多全实数域F上的许多模椭圆曲线(如果可能，也包括交换曲面)的BSD猜想的完全验证。我们将要开发的算法和将得到的数据也将在本项目的框架之外有用。

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.