The refined Coates-Sinnott conjecture for characteristic \(p\) global fields (Q1946722)

The authors aim to prove a refined function field analogue of the Coates-Sinnott conjecture, which was originally formulated in the number field context. Let us recall the setting : \(K/k\) is an abelian finite extension of global fields of characteristic \(p \not=0,\) \(G = \mathrm{Gal}(K/k),\) \(S\) is a finite non empty set of primes of \(K\) containing all those primes which ramify in \(K/k,\) \({\mathcal O}_{K, S}\) the ring of \(S\)-integers of \(K,\) \(\Theta_{K/k, S}\;:\;{\mathbb C} \to {\mathbb C} [G]\) the \(S\)-incomplete \(G\)-equivariant \(L\)-function for \(K/k,\) \(q\) the order of the exact field of constants of \(k.\) Then the authors show that, for any integer \(n \geq 2\) : \[ \mathrm{Fit}_{{\mathbb Z}[G]} \Bigl( K_{2n-1} \bigl( {\mathcal O}_{K, S}\bigl) \Bigl) \ldotp \Theta_{K/k, S} (q^{n-1}) = \mathrm{Fit}_{{\mathbb Z} [G]} \Bigl( K_{2n-2} \bigl( {\mathcal O}_{K, S} \bigl) \Bigl) \] (the original conjecture only stated an inclusion of annihilator ideals). The proof proceeds \(\ell\)-adically, i.e. \(\ell\)-component by \(\ell\)-component. First, the Quillen-Lichtenbaum conjecture (now a theorem) allows to replace the \(K\)-groups by étale cohomology groups. Second, Iwasawa theory enters the game via an (abelian) Equivariant Main Conjecture (EMC for short) recently proved by \textit{C. Greither} and \textit{C. D. Popescu} [Int. Math. Res. Not. 2012, No. 5, 986--1036 (2012; Zbl 1254.11063)] in terms of a certain ``Picard 1-motif'' \({\mathcal M}_{S, T} \;:\) let \(T\) be a finite set of primes of \(k\) such that \(S \cap T = \emptyset\) ; denoting by \(\Theta_{S, T} (q^{-s})\;:\;{\mathbb C} \to {\mathbb C} [G]\) a suitably modified equivariant \(L\)-function (which is \(\Theta_S\) if \(T = \emptyset),\) there exists a 1-motif \({\mathcal M}_{S, T}\) (whose precise definition we do not recall) such that \smallskip\noindent (1) \(T_\ell \bigl( {\mathcal M}_{S, T}\bigl)\) is \({\mathbb Z}_\ell\) \([H]\)-projective \smallskip\noindent (2) \(\mathrm{Fit}_{{\mathbb Z}_\ell [[{\mathcal G}]]} \bigl( T_\ell ({\mathcal M}_{S, T}) \bigl)\) is generated by \(\Theta_{S, T} (\gamma_q^{-1})\) (abelian EMC). \smallskip\noindent Here, for the compositum \(\widehat K\) (resp \(\widehat k)\) of \(K\) (resp. \(k)\) and an algebraic closure of \({\mathbb F}_p,\) \({\mathcal G}\) denotes \(\mathrm{Gal} (\widehat K/k)\) and \(H\) denotes \(\mathrm{Gal}(\widehat K/ \widehat k)\) ; \(\gamma_q\) is the \(q\)-power Frobenius. A (non trivial) descent-codescent via \(\Gamma = \mathrm{Gal}(\widehat K/K)\) gives that \(\mathrm{Fit}_{{\mathbb Z}_\ell[G]} H^1_{\text{\sevenrm ét}} \bigl( {\mathcal O}_{K, S}, {\mathbb Z}_\ell (n) \bigl) \ldotp \Theta_{K/k, S} (q^{n-1}) = \mathrm{Fit}_{{\mathbb Z}_\ell [G]} H^2_{\text{\sevenrm ét}} \bigl( {\mathcal O}_{K, S}, {\mathbb Z}_\ell (n) \bigl).\) In the case where \(\ell = p,\) an additional study shows that \(\Theta_{K/k, S} (q^{n-1})\) is invertible in \({\mathbb Z}_p [G].\) The main theorem follows. Remark : Although this article deals with characteristic \(p\) global fields, its references to the case of number fields certainly need to be completed. The Coates-Sinnott conjecture away from 2 was proved by \textit{T. Nguyen Quang Do} [J. Théor. Nombres Bordx. 17, No. 2, 643--668 (2005; Zbl 1098.11054)] by using an abelian EMC due to Ritter and Weiss for \(p \not= 2\) ; the case of the prime 2 was settled by \textit{R. Taleb} [Doc. Math., J. DMV 18, 749--783 (2013; Zbl 1280.11067)], who also showed an abelian Ritter-Weiss' EMC for \(p =2\) ; a non abelian generalization was obtained by \textit{A. Nickel} [Proc. Lond. Math. Soc. (3) 106, No. 6, 1223--1247 (2013; Zbl 1273.11155)], who also showed the equivalence between Ritter-Weiss' and Greither-Popescu's EMC.