##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory Seminar

## Jishnu Ray

#### University of British Columbia

## Conjectures in Iwasawa Theory of Selmer groups and Iwasawa Algebras

##### Abstract:

The Iwasawa Theory of Selmer groups provides a natural way for p-adic approach to the celebrated Birch and Swinnerton Dyer conjecture. Over a non-commutative p-adic Lie extension, the (dual) Selmer group becomes a module over a non-commutative Iwasawa algebra and its structure can be understood by analyzing ``(left) reflexive ideals'' in the Iwasawa algebra. In this talk, we will start by recalling several existing conjectures in Iwasawa Theory and then we will give an explicit ring-theoretic presentation, by generators and relations, of such Iwasawa algebras and sketch its implications in understanding the (two-sides) reflexive ideals. Generalizing Clozel's work for $SL(2)$, we will also show that such an explicit presentation of Iwasawa algebras can be obtained for a much wider class of p-adic Lie groups viz. uniform pro-p groups and the pro-p Iwahori of $GL(n,Z_p)$. Further, if time permits, I will also sketch some of my recent Iwasawa theoretic results joint with Sujatha Ramdorai.

Host: Claus Sorensen

### April 23, 2020

### 2:00 PM

### Zoom http://www.math.ucsd.edu/$\sim$nts/

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