Special Values of Abelian L - Functions at S = 0
نویسنده
چکیده
In [12], Stark formulated his far-reaching refined conjecture on the first derivative of abelian (imprimitive) L–functions of order of vanishing r = 1 at s = 0. In [10], Rubin extended Stark’s refined conjecture to describe the r-th derivative of abelian (imprimitive) L-functions of order of vanishing r at s = 0, for arbitrary values r. However, in both Stark’s and Rubin’s setups, the order of vanishing is imposed upon the imprimitive L–functions in question somewhat artificially, by requiring that the Euler factors corresponding to r distinct completely split primes have been removed from the Euler product expressions of these L–functions. In this paper, we formulate and provide evidence in support of a conjecture in the spirit of and extending the Rubin-Stark Conjectures to the most general (abelian) setting: arbitrary order of vanishing abelian imprimitive L-functions, regardless of their type of imprimitivity. The second author’s conversations with Harold Stark and David Dummit (especially regarding the order of vanishing 1 setting) were instrumental in formulating this generalization.
منابع مشابه
Degeneration of polylogarithms and special values of L-functions for totally real fields
The degeneration of the polylogarithm on the universal abelian scheme over a Hilbert modular variety at the boundary is described in terms of (critical) special values of the L-function of the totally real field defining the variety. This gives a relation between the polylogarithm on abelian schemes and special values of L-functions. 2000 Mathematics Subject Classification: 11F41, 11G55, 11R42
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