On various generalizations of Iwasawa theory

我们得到了关于广义Iwasawa理论的几个结果，特别是关于Iwasawa理论在正特征的全局场上研究阿贝尔变分的函数场模拟和Iwasawa理论的变形理论推广。对于Iwasawa理论的代数面，我们证明了控制定理，研究了Hilbert模形式的Hida变形上的Selmer群的行为。对于理论的解析方面，我们在Hilbert模形式的Hida变形上构造了若干变量的p进l函数。我们还研究了西格尔模形式的Hida变形。我们建造了科尔曼号。Perrin-Riou图，在变形上插入双指数图。我们还推广了变形空间的欧拉系统上的结果，消除了变形环是规则的假设。

We obtained several results on generalized Iwasawa theory especially on the function-field analogue of Iwasawa theory studying abelian varieties over global fields of positive characteristic and the deformation-theoretic generalization of Iwasawa theory. For the algebraic side of Iwasawa theory, we proved the control theorem to study the behavior of Selmer groups on Hida deformations of Hilbert modular forms. For the analytic side of the theory, we constructed the p-adic L-functions in several variables on Hida deformations of Hilbert modular forms. We also worked on the Hida deformations of Siegel modular forms. We constructed the Coleman. Perrin-Riou map which interpolates dual exponential maps on the deformation. We also generalized our result on Euler system for deformation spaces removing the assumption that the deformation ring is regular.