One of the programs of Stark’s
conjectures is to find as many connections as possible between the values that Artin
L–functions or their derivatives take (especially at s = 0) and arithmetic
information associated to algebraic number fields. The most refined of Stark’s
conjectures involves the values of first derivatives of L–functions at s = 0. It was
recognized early on that the conjecture should be extended to cover cases
where the order of vanishing of the L–functions at s = 0 is greater than one.
In 1980, Stark posed a question along these lines that we will consider in
detail here. In particular, we will study his question for relative quadratic
extensions and prove that an affirmative answer to his question exists for all cases
considered.
