Explicit Approaches to Modular Forms and Modular Abelian Varieties

为了获得Stein的DMS-0400386奖，Birch和Swinnerton-Dyer猜想以及Mazur关于Shafarevich-Tate群的可见性的概念激发了本研究项目的计算和理论目标。 计算目标是发展新的算法、表格和软件来研究模形式和模交换簇。 PI希望开发新的计算工具，包括一个主要的新软件包，用于计算数域上的模交换簇，并增强模形式数据库，该数据库被许多研究模形式的数学家使用。 理论目标是证明椭圆曲线的Mordell-Weil群和Shafarevich-Tate群与交换簇之间的新定理。 这些对Mazur的可见性概念的研究，以及它如何将Mordell-Weil和Shafarevich-Tate群联系起来，可能会为Birch和Swinnerton-Dyer猜想的不同情况之间的关系提供新的见解。椭圆曲线和模形式在现代数论和算术几何中发挥着核心作用。 例如，安德鲁·怀尔斯证明了费马大定理，证明了由格哈德·弗雷附加到费马主张的一个反例上的椭圆曲线将附加到一个模形式上，而这个模形式不存在。 我们对椭圆曲线和模形式世界的理解是广泛的，但许多问题仍未得到解决。这个项目的第一个目标是提供理论和计算工具，使模形式和与之相连的对象非常明确，这样数学家就可以用它们来计算，测试他们的数学，并对它们有更好的感觉。 第二个目标是利用这些工具和其他思想来更深入地理解Bryan Birch和Peter Swinnerton-Dyer关于椭圆曲线算术的猜想。

ABSTRACT for the award DMS-0400386 of Stein The Birch and Swinnerton-Dyer conjecture and Mazur's notion ofvisibility of Shafarevich-Tate groups motivate the computational andtheoretical goals of this research project. The computational goalsare to develop new algorithms, tables, and software for studyingmodular forms and modular abelian varieties. The PI hopes to createnew computational tools, including a major new package for computingwith modular abelian varieties over number fields, and enhance themodular forms database, which is used by many mathematicians who studymodular forms. The theoretical goals are to prove new theorems thatrelate Mordell-Weil and Shafarevich-Tate groups of elliptic curves andabelian varieties. These investigations into Mazur's notion ofvisibility, and how it links Mordell-Weil and Shafarevich-Tate groups,may provide new insight into relationships between different cases ofthe Birch and Swinnerton-Dyer conjecture.Elliptic curves and modular forms play a central role in modern numbertheory and arithmetic geometry. For example, Andrew Wiles provedFermat's Last Theorem by showing that the elliptic curve attached byGerhard Frey to a counterexample to Fermat's claim would be attachedto a modular form, and that this modular form cannot exist. Ourunderstanding of the world of elliptic curves and modular forms isextensive, but many questions remain unresolved. The first goal ofthis project is to provide theoretical and computational tools to makemodular forms and objects attached to them very explicit, so thatmathematicians can compute with them, test their conjectures onthem, and gain a better feeling for them. The second goal is to usethese tools and other ideas to gain a deeper understanding of theconjecture of Bryan Birch and Peter Swinnerton-Dyer about the arithmetic of elliptic curves.