##### Department of Mathematics,

University of California San Diego

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### Special Colloquium

## Cristian Popescu

#### Johns Hopkins University

## Stark-type Conjectures "over Z"

##### Abstract:

In the 1970s and early 1980s Stark developed a remarkableconjecture aimed at interpreting the first non-vanishing derivative of anArtin L-function $L_{K/k, S}(s, chi)$ at $s=0$ in terms of arithmeticproperties of the Galois extension of global fields K/k. Work of Tate,Chinburg, and Stark himself has revealed far reaching applications ofStark's Conjecture to Hilbert's 12-th Problem and the theory of Galoismodule structure of groups of units and ideal-class groups. In his searchfor new examples of Euler Systems, Rubin has formulated in 1994 a strongversion ("over Z", in Tate's terminology) of Stark's Conjecture forabelian L-functions of arbitrary order of vanishing at s=0. Our study ofthe functorial base-change behavior of Rubin's Conjecture led us toformulating a seemingly more natural Stark-type conjecture "over Z". Wewill discuss and provide evidence for this new statement, as well asbriefly describe the main goals of the conjectural program initiated byStark.

Host: Harold Stark

### February 28, 2003

### 1:00 PM

### AP&M 6438

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