##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory

## Cristian D. Popescu

#### UCSD

## On the Coates-Sinnott-Lichtenbaum Conjectures -- Quillen $K$-theory and special values of $L$-functions

##### Abstract:

The conjectures in the title were formulated in the late 1970s as vast generalizations of the classical theorem of Stickelberger. They make a very subtle connection between the $\Bbb Z[G(F/k)]$--module structure of the Quillen K-groups K${_\ast}(O_F)$ in an abelian extension $F/k$ of number fields and the values at negative integers of the associated $G(F/k)$--equivariant $L$--functions $\Theta_{F/k}(s)$. These conjectures are known to hold true if the base field $k$ is $\Bbb Q$, due to work of Coates-Sinnott and Kurihara. In this talk, we will provide evidence in support of these conjectures over arbitrary totally real number fields $k$.

### December 6, 2007

### 1:00 PM

### AP&M 7321

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