Congruence-torsion subgroups of dimension one formal groups
Let \(K\) be a finite extension of \(\mathbb{Q}_p\), with valuation ring \(R = {\mathfrak o}_K\) and let \(\overline K\) be an algebraic closure of \(K\), with valuation ring \({\mathfrak o}_{\overline K}\). Denote by \(e_0\) the ramification index of \(K\) over \(\mathbb{Q}_p\) and suppose that the valuation of \(K\) is normalized by setting \(v(p) = e_0\). Let \(G = \text{Spf} R X\) be a one-dimensional (commutative) formal group of height \(h\) and let \(\Gamma\) be an (abstract) finite abelian group with exponent dividing \(p^m\) and with \(p\)-rank less than or equal to \(h\). For such a group \(\Gamma\), define the Weyr invariants as the integers \(\gamma_i : = \dim_{\mathbb{F}_p} {p^{i - 1} \Gamma \over p^i \Gamma}\) for \(i = 1, \dots, m\). Under the above hypotheses, \(\Gamma \subset G ({\mathfrak o}_{\overline K})_{\text{tors}} \cong (\mathbb{Q}_p/ \mathbb{Z}_p)^h\), and a theorem of \textit{J. Lubin} says that there is a \({\mathfrak o}_{\overline K}\)-bialgebra \(B\), isomorphic to a quotient of \({\mathfrak o}_{\overline K} X\), such that \(\text{Spf} B ({\mathfrak o}_{\overline K}) = \Gamma\). A natural problem is to determine a finite extension of \(\mathbb{Q}_p\) and a bialgebra over the integers of such extension having \(\Gamma\) as group of \({\mathfrak o}_{\overline K}\)-points. The authors look for a lower bound for the ramification \(e_0\) in order to realize \(\Gamma\) as a congruence subgroup of a given one-dimensional formal \(R\)-group \(G\), where a congruence subgroup of \(G\) is the subset of \(\ker [p^m]_G\) consisting of the points of valuation \(\geq C\), for some \(C\). A result of \textit{J. Lubin} [Trans. Am. Math. Soc. 251, 103-127 (1979; Zbl 0431.14014)] describes the bialgebras of the subgroups in terms of the Newton polygon of the series \([p^m] (X)\). The authors prove that, for a given \(\Gamma\), there exists a congruence subgroup \(H = \text{Spf} B\) of a one-dimensional formal group \(G\), both defined over \(R\), such that \(H(K)\) has the same Weyr invariants as \(\Gamma\), if the ramification index \(e_0\) satisfies the inequality \(e_0 \geq 2 p^{\mu_e - \gamma_1} \left( 1 + {2 \over p} \right)^w\), where \(\mu_e = \text{rank} B\) and \(w\) is the number of distinct \(\gamma_i \). Moreover, they prove that the number of isomorphism classes of congruence subgroups over \(R\), having the same Weyr invariants as \(\Gamma\), is greater than or equal to \(ke^z_0\), where the rational number \(k > 0\) and the integer \(z \geq 1\) depend only on \(\Gamma\).
scientific article; zbMATH DE number 705011