Isogenies of polynomial formal groups
A polynomial formal group is a commutative \(n\) dimensional formal group law \(F(x, y) = (F_1(x, y),\dots , F_n(x, y))\) such that the power series in \(n\) variables \(F_i(x, y)\) are polynomials. For \(n=1\), examples are \(F(x, y) = x+y + cxy\). For \(n>1\) see \textit{L. Childs, C. Greither, D. J. Moss, J. Sauerberg} and \textit{K. Zimmermann}, ``Hopf algebras, polynomial formal groups, and Raynaud orders'', Mem. Am. Math. Soc. 136, No. 651 (1998)]. In the very interesting paper under review, the author examines the following question: Which of a particular family of Hopf orders \(H_\nu(i, j)\) [defined by \textit{C. Greither}, Math. Z. 210, No.~1, 37-68 (1992; Zbl 0737.11038); see also \textit{R. Larson}, J. Algebra 38, 414-452 (1976; Zbl 0407.20007)] in the group ring of a cyclic group of order \(p^2\) can be realized as a kernel of an isogeny between polynomial formal groups? Here the field \(K\) is a \(p\)-adic field and \(H_\nu(i, j)\) yields a finite flat group scheme over the ring of integers of \(K\). The question is motivated by the Tate-Oort-Raynaud classification of finite flat group schemes of type \((p,\dots , p)\), and by the fact that a realization using polynomial formal groups yields an explicit simple description of the principal homogeneous spaces for the group scheme. The latter is interesting in the context of Galois module structure. See also the introduction of the book ``Hopf algebras, polynomial formal groups and Raynaud orders'', cited above. The method used by the author is that of a resolution \(\text{Spec}(H_\nu(i,j))\rightarrow \text{Spec}(B)\rightarrow\text{Spec}(C)\) in the flat topology, which over \(K\) is just \(\mu_{p^2} \rightarrow G_m^2 \rightarrow G_m^2\). Under certain conditions on the order \(H_\nu(i, j)\) in terms of the ramification index of \(K\), the completions of the schemes \(\text{Spec}(B)\) and \(\text{Spec}(C)\) at their augmentation ideals become two-dimensional formal groups and \(\text{Spec}(H_\nu(i, j))\) is thus exhibited as a kernel of an isogeny. We remark that not every two dimensional polynomial formal group is of the form considered by the author (see theorem 2.0).
scientific article; zbMATH DE number 1271815