Description
Abstract:The Selmer group of an elliptic curve over a number field encodes several arithmetic data of the curve providing a p-adic approach to the Birch and Swinnerton-Dyer, connecting it with the p-adic L-function via the Iwasawa main conjecture. Under suitable extensions of the number field, the dual Selmer becomes a module over the Iwasawa algebra of a certain compact p-adic Lie group over Z_p (the ring of p-adic integers), which is nothing but a completed group algebra. The structure theorem of GL(2) Iwasawa theory by Coates, Schneider and Sujatha (C-S-S) then connects the dual Selmer with the “reflexive ideals” in the Iwasawa algebra. We will give an explicit ring-theoretic presentation, by generators and relations, of such Iwasawa algebras and sketch its implications to the structure theorem of C-S-S. Furthermore, such an explicit presentation of Iwasawa algebras can be obtained for a much wider class of p-adic Lie groups viz. pro-p uniform groups and the pro-p Iwahori of GL(n,Z_p). Alongside Iwasawa theoretic results, we will state results counting the dimension of first cohomology group of the pro-p Iwahori subgroup of any reductive group over Z_p and thus prove the Inverse Galois problem for p-adic Lie extensions. We finally conclude by connecting GL(2) Iwasawa theory of (C-S-S) with PGL(2) Iwasawa theory, thus moving down the Iwasawa theoretic tower, unlike (C-S-S) where their arguments circles on moving up the Iwasawa theoretic tower.
