POWRE: Abelian Varieties and Shimura Varieties

研究方向是数论和算术代数几何，重点是阿贝尔变分和志村变分。所要研究的问题涉及到f进表示、阿德尔表示、椭圆曲线的秩、定义域、模域和阿贝尔变体上的扭转点。Silverberg将使用power基金在加州大学伯克利分校担任客座教授。这对她的研究、职业发展、知名度和专业成长都有积极的帮助。阿贝尔变数和志村变数是重要的数学对象，它们出现在许多领域，并被用于Faltings对莫德尔猜想的证明和Wiles对半稳定志村谷山猜想的证明。算术代数几何是一门学科，融合了数学的两个最古老的领域：数论和代数几何。事实证明，这种结合非常富有成效，最近解决了几代人经受住的问题。在许多应用中有新的纠错码。这些代码对现代计算机（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技术中有许多应用。

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.