p-adic Iwasawa theory for Galois representations

Iwasawa理论是数论的一个分支，它的基本问题将算术问题（如多项式方程的有理解）与解析信息（如某些复解析函数的值，即所谓的l函数）联系起来。这两种截然不同的思想的融合使岩泽理论成为一个神秘而强大的领域。通过它的主要猜想，岩泽理论仍然是今天已知的唯一系统的方法，用于研究算术问题和l函数的特殊值之间的深刻联系，典型的是Birch和Swinnerton-Dyer猜想。椭圆曲线是一类二元多项式方程，可以用Iwasawa理论来研究椭圆曲线的算法。关于椭圆曲线的算术问题与数论中一些最古老的问题有关，例如与具有某些性质的直角三角形是否存在有关的同余数问题。椭圆曲线在怀尔斯证明费马大定理的过程中也发挥了核心作用。在提议的研究中，我计划使用最近从另一个称为“p进分析”的数学领域开发的工具来研究Iwasawa椭圆曲线理论的某些方面。这将对代数数论中的一些重要问题（如Birch- Swinnerton-Dyer猜想）提供新的见解。

Iwasawa theory is a subfield of number theory whose fundamental questions relate arithmetic problems (e.g. rational solutions to polynomial equations in several variables) to analytic information (e.g. values of certain complex-analytic functions, so-called L-functions). The merging of these two disparate ideas makes Iwasawa theory a mysterious and powerful field. Via its main conjectures, Iwasawa theory remains the only systematic method known today for studying the deep connections between the arithmetic problems and special values of L-functions, typified by the conjecture of Birch and Swinnerton-Dyer. One can use Iwasawa theory to study the arithmetic of elliptic curves, which are a class of polynomial equations in two variables. Questions about the arithmetic of elliptic curves are linked to some of the oldest problems in number theory, such as the congruent number problem which concerns the existence of right-angled triangles with certain properties. Elliptic curves also played a central role in Wiles' proof of Fermat's last theorem. In the proposed research I plan to investigate some aspects of the Iwasawa theory of elliptic curves using recently developed tools from an another area of mathematics called 'p-adic analysis'. This should give new insight into some important problems (such as the Birch--Swinnerton-Dyer conjecture) in algebraic number theory.