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.

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.

Description

Ramanujan Hall, Department of Mathematics

Date

Fri, August 2, 2019

Start Time

2:30pm-3:30pm IST

Duration

1 hour

Priority

5-Medium

Access

Public

Created by

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Updated

Mon, July 29, 2019 5:46pm IST