On various generalizations of Iwasawa theory

我们在广义Iwasawa理论，特别是Iwasawa理论的函数场类似物，正特征整体域上的阿贝尔簇以及Iwasawa理论的形变论推广等方面得到了一些结果。在岩泽理论的代数方面，我们证明了控制定理，从而研究了塞尔默群在Hilbert模形式的希达变形上的行为.在理论的解析方面，我们构造了Hilbert模形式的希达变形上的多元p-adic L-函数。我们还研究了Siegel模形式的希达变形。我们建造了科尔曼。Perrin-Riou映射，它在变形上插值对偶指数映射。在去掉变形环是正则环的假设下，我们还推广了变形空间中Euler系统的结果。

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.