Integral representations of unramified Galois groups and matrix divisors over number fields (Q1911579)

Let \(K\) be an algebraic number field with ring of integers \(\mathcal O\), \(\mathbf A\) its adele ring, \(U_n(\mathbf A)\) the maximal compact subgroup of \(GL_n(\mathbf A)\) and \(U_n(\mathcal O)=GL_n(K)\cap U_n(\mathbf A)\). Put \(G=\text{Gal}(\widetilde K/K)\), where \(\widetilde K\) is the maximal unramified extension of \(K\). In analogy to the ideas of \textit{A. Weil} [J. Math. Pures Appl., IX. Sér. 17, 47-87 (1938; Zbl 0018.06302)] for the study of the fundamental group of an algebraic curve, the author investigates the set \(R_n(\overline X)\) of integral unitary adelic representations \(\rho\colon G\to U_n(\mathcal O)\), where \(\overline X\) denotes the scheme \(\text{spec}(\mathcal O)\), completed at the infinite places. He constructs a map \(\overline{\Phi}_n\colon R_n(\overline X)\to Cl_n(\overline X)\) into the class group of \(n\)-dimensional matrix divisors, given by \(Cl_n(\overline X)=GL_n(K)\backslash GL_n(\mathbf A)/U_n(\mathbf A)\), and studies its properties. Finally, these ideas are demonstrated for the worked out example \(K=\mathbb{Q}(\sqrt{-155})\), in which case \(G\simeq S_3\).