Iwasawa Theory of Shimura Varieties

This paper provides a comprehensive survey of the Iwasawa theory of Shimura varieties, a central topic in modern number theory that synthesizes arithmetic algebraic geometry, automorphic forms, and Galois representations. We explore the historical development of both Iwasawa theory, initiated by Kenkichi Iwasawa's work on ideal class groups in cyclotomic towers, and the theory of Shimura varieties, which generalize modular curves and serve as moduli spaces for abelian varieties with additional structures. The core of the paper is dedicated to the formulation and study of the Iwasawa main conjecture in this context. This conjecture posits a deep relationship between an analytic object, a p-adic L-function constructed from automorphic data, and an algebraic object, the characteristic ideal of a Selmer group defined via Galois cohomology. We discuss the key ingredients in this framework, including the construction of Galois representations attached to automorphic forms, the role of Hida families in p-adically interpolating automorphic forms, and the use of cohomological methods to define Selmer groups. The paper reviews significant results, such as the proof of the main conjecture for GL(2) by Skinner and Urban, and its implications for the Birch and Swinnerton-Dyer conjecture. Furthermore, we examine the connection between Euler systems and the algebraic side of the main conjecture, highlighting their role in bounding Selmer groups. The methodology combines techniques from p-adic Hodge theory, deformation theory of Galois representations, and the geometry of Shimura varieties at p-adic levels. We conclude by discussing open problems and future directions, such as generalizing the main conjecture to higher-rank groups and its relationship with the p-adic Langlands program.