POWRE: Abelian Varieties and Shimura Varieties

西尔弗伯格9805854研究领域是数论和算术代数几何，重点是阿贝尔变种和下村变种。所要研究的问题涉及f-adid表示、adelic表示、椭圆曲线的阶、定义域、模域和阿贝尔簇上的扭点。西尔弗伯格将使用POWRE补助金成为加州大学伯克利分校的客座教授。这将对她的研究、职业发展、知名度和职业发展做出积极贡献。Abelian簇和Shimura簇是出现在许多领域的重要数学对象，并被用于Faltings证明Mordell猜想和Wiles证明半稳定ShimuraTaniyama猜想。算术代数几何是一门融合了数论和代数几何这两个最古老的数学领域的学科。事实证明，这种结合非常有成效，最近解决了几代人经受住的问题。在众多应用中，有一种是新的纠错码。这种代码对于现代计算机(硬盘)和光盘都是必不可少的。数论和代数几何在数据传输和处理、通信系统、物理、理论计算机科学和机器人学中有大量的应用。

Silverberg 9805854 The research is in the fields of Number Theory and Arithmetic Algebraic Geometry, with a focus on abelian varieties and Shimura varieties. The questions to be studied concern f-adic representations, adelic representations, ranks of elliptic curves, fields of definition, fields of moduli, and torsion points on abelian varieties. Silverberg will use a POWRE grant to be a Visiting Professor at the University of California at Berkeley. This will contribute positively to her research, career advancement, visibility, and professional growth. Abelian varieties and Shimura varieties are important mathematical objects which arise in a wide variety of areas, and were used in Faltings' proof of the Mordell Conjecture and in Wiles' proof of the Semistable ShimuraTaniyama Conjecture. Arithmetic Algebraic Geometry is a subject that blends two of the oldest areas of Mathematics: Number Theory and Algebraic Geometry. This combination has proved extraordinarily fruitful, having recently solved problems that withstood generations. Among the many applications are new error correcting codes. Such codes are essential for both modern computers (hard disks) and compact disks. Number Theory and Algebraic Geometry have found numerous applications in data transmission and processing, communication systems, physics, theoretical computer science, and robotics.