Executive Summary : | The interplay of arithmetic and analytic objects is a beautiful aspect of number theory, as seen in the Birch and Swinnerton-Dyer (BSD) Conjecture, which predicts a deep relation between the Hasse-Weil complex L-function of an elliptic curve and the Mordell-Weil group. Iwasawa theory, originating from Iwasawa's study of Ideal class groups in the tower of cyclotomic fields, focuses on conjectures similar to the BSD Conjecture in the p-adic world. Recent developments in higher dimensional Iwasawa theory include the "two variable" Iwasawa theory for the Hida family, the "non-commutative" Iwasawa theory, and the Rankin-Selberg product of two modular forms. The project aims to examine the pseudonullity for the dual p-infinity fine Selmer group of an elliptic curve over certain Z_p^d extensions of number fields, study the injectivity of map induced by specialization at infinitely many height 1 prime ideals of the localized Iwasawa algebra of a compact p-adic Lie group, and study the p-infinity Selmer group over false Tate curve extension and the variation under congruence for the Iwasawa main conjecture over cyclotomic Z_p extension of a number field. These studies aim to shed light on Greenberg's pseudonullity conjecture, Iwasawa's mu=0 conjecture, equivalence between the Iwasawa main conjecture for Hida family and infinitely many specilization in non-commutative Iwasawa theory, and the structure of Selmer group and Iwasawa main conjecture for Rankin-Selberg over a general number field. The theory of elliptic curves has applications in cryptography and cybersecurity. |