On the conjecture of Lichtenbaum and of Chinburg over function fields (Q1113955)

In this paper we study function field analogs of two conjectures on Artin L-functions. Let C be a smooth projective curve over a finite field, S a finite set of places of C, and V be a representation of \(G=Gal(C'/C)\) where C' is a branched Galois covering of C. Denote by \(L_ S(V,1)\) the leading coefficient of Artin L-function of the curve \(C\setminus S\) at \(q^{-s}=t=1\). Then we prove that if \(V=M\otimes_{{\mathfrak O}_ E}{\mathbb{C}}\) for some \({\mathfrak O}_ E\)-module M where \({\mathfrak O}_ E\) is the ring of integers in a number field E, \(L_ S(V,1){\mathfrak O}_ E\) can be expressed as Euler characteristic of the étale cohomology of the sheaf associated to M, which is conjectured by Lichtenbaum. Using this we prove that \(L_ S(V,1)\) modulo some regulator generates an ideal expressed by Galois cohomology, which is conjectured by Chinburg.