Description
Number theory seminar.
Speaker: Jishnu Ray.
Affiliation: University of British Columbia, Vancouver.
Date and Time: Friday 2 August, 2:30 pm - 3:30 pm.
Venue: Ramanujan Hall, Department of Mathematics.
Title: Selmer group of elliptic curves and explicit presentation of
Iwasawa algebras.
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 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.