Formal groups and Dirichlet \(L\)-functions. I
Let \(\chi\) be the non-trivial quadratic character corresponding to the quadratic number field \(K\) of discriminant \(D\), and let \(\hat H_ \chi\) be the one-dimensional formal group over \(K\) with logarithm \(\lambda(T)=\sum_{n\geq 1}{\chi(n)\over n}T^ n\). By a theorem of \textit{T. Honda} [Osaka J. Math. 5, 199--213 (1968; Zbl 0169.37601)], \(\hat H_ \chi\) is strictly isomorphic over the ring of integers of \(K\) to the one- dimensional formal group \(\hat F_ \chi\) with the group law \(F_ \chi(X,Y)=X+Y+\sqrt DXY\) defined over \(K\). The authors generalise this result to non-quadratic characters of \((\mathbb{Z}/m\mathbb{Z})^*\) with values in \(\text{GL}(n_ p,\mathbb{Z}_ p)\), where \(\mathbb{Z}_ p\) denotes the ring of \(p\)-adic integers and \(n_ p\) depends explicitly on \(p\), by constructing higher dimensional analogues of Honda's groups over \(\mathbb{Q}_ p(\zeta)\), where \(\zeta\) is a primitive \(m\)th root of unity.
higher dimensional analogues of Honda's groups
scientific article; zbMATH DE number 96564