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Rubin, Karl
A Stark conjecture “over ${\bf Z}$” for abelian $L$-functions with multiple zeros. Annales de l'institut Fourier, 46 no. 1 (1996), p. 33-62
Full text djvu | pdf | Reviews MR 97d:11174 | Zbl 0834.11044 | 3 citations in Numdam

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Abstract

Suppose $K/k$ is an abelian extension of number fields. Stark's conjecture predicts, under suitable hypotheses, the existence of a global unit $\varepsilon$ of $K$ such that the special values $L'(\chi,0)$ for all characters $\chi$ of ${\rm Gal}\slash(K/k)$ can be expressed as simple linear combinations of the logarithms of the different absolute values of $\varepsilon$.

In this paper we formulate an extension of this conjecture, to attempt to understand the values $L^{(r)}(\chi,0)$ when the order of vanishing $r$ may be greater than one. This conjecture no longer predicts the existence of individual special global units, but rather of special elements in an exterior power of the Galois module of global units (or $S$-units). We also discuss connections between this conjecture, class number formulas, and Euler systems.

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