Stark’s Question on Special Values of $L$\textendash Functions

In 1980, Stark posed a far reaching question regarding the second derivatives at $s=0$ of $L$–functions of order of vanishing two associated to abelian extensions of number fields (global fields of characteristic $0$) (see \cite{St2}). Over the years, various mathematicians, most notably Grant \cite{Gra}, Sands \cite{Sa}, and Tangedal \cite{Tan} have provided evidence in favor of an affirmative answer to Stark’s question. In \cite{P2}, we extrapolated Stark’s question to the appropriate class of $L$–functions associated to abelian extensions of function fields (global fields of characteristic $p >0$) and showed that, in general, it has a negative answer in that context. Unfortunately, the methods developed in \cite{P2} are specific to the geometric, characteristic $p >0 $ situation and cannot be carried over to the characteristic $0$ case. In the present paper, we develop new methods which permit us to prove that, in general, (even a weak form of) Stark’s question has a negative answer in its original, characteristic $0$ formulation as well.

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Published 1 January 2007