##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory

## Samit Dasgupta

#### Harvard University

## Shintani zeta-functions and Gross-Stark units for totally real fields

##### Abstract:

Let $F$ be a totally real number field and let $p$ be a finite prime of $F$, such that $p$ splits completely in the finite abelian extension $H$ of $F$. Stark has proposed a conjecture stating the existence of a $p-unit$ in $F$ with absolute values at the places above $p$ specified in terms of the values at zero of the partial zeta functions associated to $H/F$. Gross proposed a refinement of Stark's conjecture which gives a conjectural formula for the image of Stark's unit in $F_p^\times/ \widehat E$, where $F_p$ denotes the completion of $F$ at $p$ and $\widehat E$ denotes the topological closure of the group of totally positive units of $F$. We propose a further refinement of Gross' conjecture by proposing a conjectural formula for the exact value of Stark's unit in $F_p^\times$.

Host: Cristian Popescu

### October 26, 2006

### 2:00 PM

### AP&M 7321

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