Hermitian modules in Galois extensions of number fields and Adams operations (Q1185526)

For a Galois extension \(N/K\) of number fields with Galois group \(\Gamma\), let \[ t_{N/K}(x,y)=\sum_{\gamma\in\Gamma}\text{trace}_{N/K}(xy^ \gamma)\gamma^{-1} \] denote the Hermitian form on \(N\), associated with the trace function. Let \(m_ \Gamma(x,y)=\bar x y\) denote the so called multiplication form on \(K\Gamma\) where \(\bar x\) is the image of \(x\) under the canonical involution of \(K\Gamma\). Furthermore, let \(A(N/K)\) denote the square root of the inverse different of \(N/K\) and \(K_ 0H(\mathbb{Z}\Gamma)\) denote the Grothendieck group of Hermitian \(\mathbb{Z}\Gamma\)-modules. For any number field \(F\), \({\mathcal O}_ F\) will denote the ring of integers of \(F\). By restricting \(t_{N/K}\) to \({\mathcal O}_ N\) and \(A(N/K)\) two \({\mathcal O_ k\Gamma}\) Hermitian modules are obtained which are denoted by \({\mathcal O}(N/K)\) and \(A(N/K)\). The main result is the determination of a right \({\mathcal O}_ K\Gamma\)-ideal \(M=M(N/K)\) such that: (1) If \(N/K\) is tamely ramified then \(A(N/K)\) and \(({\mathcal O}_ k\Gamma,m_ \Gamma)\) define the same class in \(K_ 0 H(\mathbb{Z}\Gamma)\). (2) If \(N/K\) is domestic then \({\mathcal O}(N/K)\) and \((M,m_ \Gamma)\) define the same class in \(K_ 0 H(\mathbb{Z}\Gamma)\). An extension \(N/K\) is called domestic if all prime divisors of \([N:K]\) are unramified in \(N/K\).