##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory

## James Barrett

#### King's College London

## Dirichlet L-functions at strictly positive integers and Fitting invariants of K-groups

##### Abstract:

For K a C.M. abelian extension of a totally real base-field k with Galois group G, Solomon has recently constructed for each prime p a Z p[G] ideal of Q p[G] related to values of Dirichlet L-functions at s=1 and conjectured that this ideal is contained within Z p[G]. Jones has subsequently shown that for each odd p the Equivariant Tamagawa Number Conjecture (or ETNC) implies that Solomon's ideal should actually be contained within the Fitting ideal of the class-group of O K. I shall explain how to define analogous ideals related to values of Dirichlet L-functions at integers r strictly greater than 1 and provide a sketch of the techniques used to show that the ETNC relates these `higher Solomon ideals' to the Fitting ideals of certain natural cohomology groups (and thus, when the Quillen-Lichtenbaum conjecture is valid, to Fitting ideals of Quillen K-groups of O K). In particular, for certain choices of K/k and r these results are unconditional as the relevant cases of the ETNC and Quillen-Lichtenbaum conjecture are known to be valid.

Host: Cristian Popescu

### February 22, 2007

### 1:00 PM

### AP&M 7321

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