##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory Seminar

## Rusiru Gambheera Arachchige

#### UCSD

## An unconditional equivariant main conjecture in Iwasawa theory

##### Abstract:

In 2015 Greither and Popescu constructed a new class of Iwasawa modules, which are the number field analogues of $p-$adic realizations of Picard 1- motives constructed by Deligne. They proved an equivariant main conjecture by computing the Fitting ideal of these new modules over the appropriate profinite group ring. This is an integral, equivariant refinement of Wiles' classical main conjecture. As a consequence they proved a refinement of the Brumer-Stark conjecture away from 2. All of the above was proved under the assumption that the relevant prime $p$ is odd and that the appropriate classical Iwasawa $\mu$â€“invariants vanish. Recently, Dasgupta and Kakde proved the Brumer-Stark conjecture, away from 2, unconditionally, using a generalization of Ribet's method. We use the Dasgupta-Kakde results to prove an unconditional equivariant main conjecture, which is a generalization of that of Greither and Popescu. As applications of our main theorem we prove a generalization of a certain case of the main result of Dasgupta-Kakde and we compute the Fitting ideal of a certain naturally arising Iwasawa module. This is joint work with Cristian Popescu.

[Pre-talk at 1:20PM]

### October 27, 2022

### 2:00 PM

APM 6402 and Zoom

See https://www.math.ucsd.edu/~nts

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