Mathematics

Dr. Jishnu Ray
The University of British Columbia, Canada.

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 Lfunction 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 padic 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). If we have time, alongside Iwasawa theoretic results, we will state results (joint with Christophe Cornut) constructing Galois representations with big image in reductive groups and thus prove the Inverse Galois problem for p-adic Lie extensions using the notion of “p-rational” number fields.