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Special values of Abelian \(L\)-functions at \(s=0\). (English) Zbl 1166.11043

The purpose of Stark’s conjectures is to extract information on arithmetic invariants of global field extensions \(K/k\) from special values of the associated Artin \(L\)-functions. H. M. Stark’s original integral conjecture [Adv. Math. 35, 197–235 (1980; Zbl 0475.12018)] predicted an arithmetic formula for the first derivative of an abelian \(S\)-imprimitive \(L\)-function at \(s=0\) under the presence in the set \(S\) of primes whose Euler factors are missing of a distinguished prime \(\nu_0\) which splits completely in \(K/k\). K. Rubin [Ann. Inst. Fourier 46, No. 1, 33–62 (1996; Zbl 0834.11044)] presented a conjecture which extended Stark’s to the \(r\)th derivative under the presence of \(r\) splitting primes in \(S\). The aim of this work is to formulate and provide evidence for 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.

MSC:

11R42 Zeta functions and \(L\)-functions of number fields
11R58 Arithmetic theory of algebraic function fields
11R27 Units and factorization
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