Arithmetic of Abelian Varieties

调查员将进行数论和算术代数Goemetry的研究，重点是椭圆曲线。 她将研究分布的行列椭圆曲线在家庭的二次扭曲，调查如何经常出现一个给定的排名。 当人们研究Birch和Swinnerton-Dyer猜想以及椭圆曲线的秩的有界性或无界性时，这些问题就出现了。椭圆曲线既有数学上的重要性，又有强大的应用。 他们在实现算术和几何的重要和基本的数学结果中至关重要，包括费马大定理，志村-谷山猜想和莫德尔猜想。 它们也可以用来分解整数和测试素数。 它们在密码学中有重要的应用，用于无线和/或移动的设备，包括智能卡、蜂窝电话、寻呼机和手持计算机。

The investigator will conduct research in Number Theory and Arithmetical Algebraic Goemetry, with a focus on elliptic curves. She will study the distribution of ranks of elliptic cuves in families of quadratic twists, investigating how often a given rank occurs. Such questions arise when one studies the Conjecture of Birch and Swinnerton-Dyer, and conjectures on the boundedness or unboundedness of ranks of elliptic curves.Elliptic curves have both mathematical importance, and powerful applications. They were crucial in achieving important and fundamental mathematical results about arithmetic and geometry, including Fermat's Last Theorem, the Shimura-Taniyama Conjecture, and the Mordell Conjecture. They can also be used to factor integers and to test for primality. They have important applications to cryptography, for use in wireless and/or mobile devices including smart cards, cell phones, pagers, and hand-held computers.