On the values of equivariant zeta functions of curves over finite fields
Let \(K/k\) be a finite Galois extension of global function fields of characteristic \(p\) whith Galois group \(G\). Let \(C_K\) be the unique geometrically irreducible smooth projective curve which has function field equal to \(K\). For each finite non-empty set \(S\) of places of \(k\) that contains all places which ramify in \(K/k\), let \({\mathcal O}_{K,S}\) be the subring of \(K\) consisting of those elements that are integral at all places of \(K\) which do not lie above an element of \(S\), and let \(U_{K,S}:=\text{Spec}({\mathcal O}_{K,S})\). The author conjectures an equality between an element of the relative algebraic \(K\)-group \(K_0({\mathbb Z}[G],{\mathbb R})\) constructed from the leading term in the Taylor expansion at \(s=0\) of the \(G\)-equivariant Artin \(L\)-function of \(U_{K,S}\) and the refined Euler characteristic of a pair comprising the Weil-étale cohomology of \({\mathbb G}_m\) on \(U_{K,S}\) and a natural logarithmic regulator mapping. He proves the \(l\)-primary component of this conjecture for all primes \(l\) for which either \(l\neq p\) or \(K_0({\mathbb Z}_l[G],{\mathbb Q}_l)\) is torsion free. He shows that this result has the following consequences for \(K/k\): if \(p\) does not divide the order of \(G\), then refined versions of all of Chinburg's ``\(\Omega\)-conjectures'' in Galois module theory are valid; if the torsion subgroup of \(K^\times\) is a cohomologically-trivial \(G\)-module, then Gross's conjectural ``refined class number formula'' is valid; if \(K/k\) satisfies a certain natural class-field theoretical condition, then Tate's refinement of Gross's conjecture is valid.
scientific article; zbMATH DE number 2146805