Actions of Galois groups on invariants of number fields (Q927711)

Let \(L/K\) be a Galois extension of number fields with Galois group \(G\). This paper explores connections between norm idempotents of subgroups \(H\) of \(G\) and arithmetic data associated to the fixed fields \(L_H\) such as the number of real and imaginary embeddings, class groups, unit groups, and Arakelov genera. This is a number field analogue of results by E. Kani and M. Rosen for Jacobians of curves over algebraically closed fields, and in some sense shows that functions related to the \(p\)-part of the class group and unit group behave like \(p\)-ranks of Jacobians of curves. More specifically, define the norm idempotent of \(H\) to be the \(K[G]\) element \[ \varepsilon_H = \frac{1}{|H|} \sum_{h \in H} h. \] Let \(\lambda_H = r_H + s_H - 1\), where \(r_H\) is the number of real embeddings of \(L_H\) and \(2s_H\) is the number of complex embeddings of \(L_H\). Let \(\lambda_{H,p,n}\) denote the number of times \(\mathbb{Z}/p^n\mathbb{Z}\) appears as a direct summand in the cyclic group decomposition of the \(p\)-part of the class group of \(L_H\), and let \(k_{H,p,n}\) denote the similar quantity for the unit group. The main result of the paper is that a linear dependence \(\sum t_H \varepsilon_H = 0\) implies the linear dependence \(\sum t_H \lambda_H = 0\). Moreover, if \(t_H = 0\) for all subgroups of order divisible by \(p\), then \(\sum t_H \epsilon_H = 0\) implies \(\sum t_H \lambda_{H,p,n} = 0\) and \(\sum t_H k_{H,p,n} = 0\). The paper goes on to establish similar results for the Arakelov genus of \(L_H\) and for another invariant \(\eta_{L_H}\) associated to \(L_H\) by considering norms on its Minkowski space.