Variation of Selmer Groups

Ramakrishna DMS-0400232奖摘要：本课题的主要目的是继续研究模p伽罗瓦表示的PI、p进伽罗瓦表示及其与数论和算术代数几何问题的关系。特别强调的是理解当更多素数加入到分支集中时，各种塞尔默群是如何变化的。在一种情况下，这种塞尔默群的变化与有多少“新在q”的新形式与特定的旧形式一致有关。PI的研究方向是代数数论。这个领域植根于对整数及其各种性质的研究。PI研究的一个关键工具是伽罗瓦理论。伽罗瓦理论是对方程解的对称性的研究。对这些对称性的理解，在基本的数学问题的解本身上，显得微不足道。这种对称性的一个例子是多项式方程的复解出现在复共轭对中。

Abstract for award DMS-0400232 of Ramakrishna: The main purpose of this project is to continue aninvestigation of the PI of mod p Galois representations,p-adic Galois representations and their relations withproblems in number theory and arithmetic algebraic geometry.A particular emphasis will be the understanding of how Selmergroups of various sorts change as more primes are addedto the ramification set. In one context this variationof Selmer groups relates to how many `new at q' newforms there arecongruent to a particular oldform. The PI's research is in Algebraic Number Theory.This field has its roots in the study of whole numbers and theirvarious properties. A key tool in the PI's researchis Galois theory. Galois theory is the study of symmetries of solutionsof equations. The understanding of these symmetriesshed slight on the the solutions themselves, abasic mathematical question. An example of such a symmetry is thefact that complex solutions to polynomial equations occur incomplex conjugate pairs.