##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory

## Cristian Popescu

#### UCSD

## 1-Motives, Etale Cohomology and Equivariant Iwasawa Theory

##### Abstract:

The classical conjectures of Gross and Brumer-Stark seem to describe two completely unrelated properties of special values of Galois equivariant global L-functions. In this talk, we will develop a general Equivariant Main Conjecture in Iwasawa Theory which captures the Brumer-Stark and Gross phenomena simultaneously and works equally well in characteristics $0$ and p. The characteristic p side of the theory draws its main ideas from Deligne\'s construction of $1-motives$ associated to smooth, projective curves over finite fields. The characteristic $0$ side of the theory is based on our new construction of number field analogues of the l-adic realizations (i.e. l-adic etale cohomology groups) of Deligne\'s $1-motives$ and is deeply rooted in earlier work of Tate and Ritter - Weiss on the theory of multiplicative Galois module structure. Time permitting, we will also provide evidence in support of this new equivariant Iwasawa theoretic statement and discuss its links to l-adic refinements of integral Rubin - Stark - type conjectures on special values of global L-functions.

Host:

### November 20, 2003

### 1:00 PM

### AP&M 7321

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