Explicit Approaches to Modular Forms and Modular Abelian Varieties

对于斯坦的DMS-0400386奖，白桦和Swinnerton-Dyer猜想和Mazur的Shafarevich-Tate群可见性的概念激发了这一研究项目的计算和理论目标。计算的目标是开发新的算法、表格和软件来研究模形式和模阿贝尔变种。PI希望创建新的计算工具，包括一个主要的新程序包，用于计算数域上的模阿贝尔变体，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和交换簇的Mordell-Weil和Shafarevich-Tate群之间的新定理。这些对Mazur可见性概念的研究，以及它如何将Mordell-Weil和Shafarevich-Tate群联系在一起，可能会为理解Birch猜想和Swinnerton-Dyer猜想之间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，Andrew Wiles证明了Fermat最后定理，证明了Gerhard Frey附在Fermat声明的反例上的椭圆曲线将附在模形式上，并且这个模形式不存在。我们对椭圆曲线和模形式的世界的理解是广泛的，但许多问题仍然没有解决。这个项目的第一个目标是提供理论和计算工具，使模形式和附着在它们上的对象非常明确，以便数学家可以用它们进行计算，检验他们的猜想，并获得对它们的更好的感觉。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.