Values of \(L\)-functions at \(s = 0\) (Q1203528)

Based on the author's introduction: Let \(V\) be a Galois representation (over a number field or a function field) and \(L_ S(s,V)\) be the function obtained by removing from the Artin \(L\)-function of \(V\) the Euler factors associated to the finite places of a finite set \(S\). One of Stark's conjectures says, very roughly speaking, that the value of \(L_ S(s,V)\) at \(s=0\) has some kind of rationality property. Two subsequent conjectures assert that this value can be expressed in terms of Artin- Verdier cohomology (Lichtenbaum) or of Galois cohomology (Chinburg). In the function field case, both conjectures are true. In fact, Lichtenbaum's conjecture can be proved by means of the \(\ell\)-adic étale and crystalline cohomological expression of \(L\)-functions (not available for number fields), and the author has shown that this conjecture implies Chinburg's conjecture [Math. Ann. 285, 417-445 (1989; Zbl 0662.14007)]. In the present paper, he reduces both conjectures to the case where the representation is faithful and one-dimensional. He then adapts this proof for the function field case to the number field case in order to show that the two conjectures are equivalent.