Trace forms and Stickelberger relations
Let \(K\) be an algebraic number field. Let \(N/K\) be an abelian extension of odd degree with Galois group \(G\). Then \(A_{N/K}\) denotes the square root of the inverse different of \(N/K\). Let \(t_{N/K} (x,y) : = \text{trace}_{N/K} (x \cdot y)\) be the trace form of \(N/K\). Then \((A_{N/K}, t_{N/K})\) is a \(G\)-equivariant symmetric bilinear form over the ring of integers \({\mathcal O}_K\) of \(K\). Let \(t_G\) be the standard symmetric bilinear form on the group algebra \(KG\). A pair \((L,b)\) consisting of an \({\mathcal O}_KG\)-lattice \(L\) and a \(G\)-invariant symmetric bilinear form \(b : L \times L \to {\mathcal O}_K\) is said to be in the principle genus if \((L,b)\) and \(({\mathcal O}_KG, t_G)\) are everywhere locally isometric as \(G\)-forms. We denote by \({\mathcal L}\) the set of \({\mathcal O}_KG\)-lattices \(L\) of \(KG\) such that \((L, t_G)\) is in the principle genus. Then \({\mathcal L}\) is a group with respect to multiplication. We define the unitary class group \(\text{UCl} ({\mathcal O}_KG)\) by \[ KG^{(1)} @>i>> {\mathcal L} @>\text{ucl}>> \text{UCl} ({\mathcal O}_KG) \to 0, \] where \(KG^{(1)}\) is the subgroup of \(KG\) of units \(u\) with \(uu^* = 1\). Further \(i(u) = u \cdot {\mathcal O}_KG\). Let \(N/K\) be a tamely ramified extension. Then \((A_{N/K}, t_{N/K})\) belongs to the principle genus and we can choose an \(G\)-equivariant isometry \(f : (N,t_{N/K}) \to (KG, t_G)\). Set \({\mathcal O}_{N/K} : = \text{ucl} (f(A_{N/K} ))\). The set \[ RU ({\mathcal O}_KG) : = \{{\mathcal O}_{N/K} |N/K \text{ is a tame } G \text{-extension}\} \] is called the realizable subset of \(\text{UCl} ({\mathcal O}_KG)\). The main result now is that for Kummer extensions of prime odd order \(l\) the realizable subset is a group. It can be characterized by a hermitian version of the Stickelberger ideal. Let \(\delta_r \in \Aut G\) be the automorphism given by \(\delta_r(g) = g^r(e \nmid r)\). Then \[ \Phi : = \sum^{(e - 1)/2}_{r \neq 0,r = {(e - 1) \over 2}}r \cdot \delta_r^{-1} \in \mathbb{Z} \Aut G \] is the hermitian Stickelberger element and \(S = {\Phi \over l} \mathbb{Z} \Aut G \cap \mathbb{Z} \Aut G\) is the hermitian Stickelberger ideal. Then \[ \begin{aligned}\nu : S \otimes_{\mathbb{Z} \Aut G} & \text{Cl} ({\mathcal O}_KG) \to \text{UCl} ({\mathcal O}_KG):\\ & \gamma \otimes \text{cl} (I) \mapsto \text{ucl} (I^\gamma)\end{aligned} \] is a well-defined homomorphism with image \(RU ({\mathcal O}_KG)\).
hermitian version of the Stickelberger ideal
scientific article; zbMATH DE number 749194