On refined Stark conjectures in the non-abelian case

We discuss an explicit integral refinement of Stark’s Conjecture in the general (non-abelian) case. We show that, upon specialization to the case of odd irreducible degree two complex characters of $\Gal(\overline{\bq}/\bq)$ for which the associated $L$-function vanishes to order one at $s=0$, our conjecture refines a question of Stark and a conjecture of Chinburg. As supporting evidence for our conjecture we give a full proof in the function field case and (in the number field case) a proof for rational valued characters and for degree one characters of either $\Gal(\overline{\bq}/\bq)$ or $\Gal(\overline{\bq}/k)$ for suitable imaginary quadratic fields $k$.

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Published 1 January 2008