Zur darstellungstheoretischen Deutung von Kohomologiegruppen (Q801970)

Let \(k\) be a local or global number field. It is well known that -- except for special cases -- the strict cohomological dimension of the absolute Galois group \(G_k= \mathrm{Gal}(\bar k/k)\) is \(2\). So for every \(n\ge 3\) and every finite Galois extension \(K/k\) with Galois group \(G\) there is a finite Galois extension \(L\) of \(K/k\) such that the inflation map \(\operatorname{inf}: H^n(G,\mathbb{Z})\to H^n(G(L/k),\mathbb{Z})\) is trivial. In this paper we ask for a reciprocity law for all the finite Galois extensions of \(K/k\) determined in this way. More precisely, we show that \(H^n(G,\mathbb{Z})\) determines canonically a set of irreducible representations \(J^n(G)\) of \(G_k\) and -- according to Langlands' philosophy -- expect that it also determines canonically a set of irreducible admissible resp. cuspidal automorphic representations \(\mathcal R^n(G)\) of \(\mathrm{GL}(m,k)\) resp. \(\mathrm{GL}(m,\mathbb{A}_k)\) -- for various \(m\) -- such that there is a canonical embedding \(J^n(G)\hookrightarrow \mathcal R^n(G)\), \(D\mapsto \pi(D)\), with the property that the Artin \(L\)-function of \(D\) coincides with the \(L\)-function of \(\pi(D)\). Using rather deep results of Kazhdan we are able to verify this in the case \(n=3\) and abelian \(G\) of type \(\mathbb{Z}/2\times \mathbb{Z}/2\) or \(\mathbb{Z}/3\times \mathbb{Z}/3\). These examples suggest that every statement in Galois cohomology should correspond to a statement in the theory of automorphic representations. For instance, the cohomological triviality of \(H^3(G_k,\mathbb{Z})\) is closely related to the problem of \(L\)-indistinguishability for \(\mathrm{SL}(m)\).