This book gives two new methods for constructing \(p\)-elementary Hopf algebra orders over the valuation ring \(R\) of a local field \(K\) containing the \(p\)-adic rational numbers. One method constructs Hopf orders using isogenies of commutative degree 2 polynomial formal groups of dimension \(n\), and is built on a systematic study of such formal group laws. The other method uses an exponential generalization of a 1992 construction of Greither. Both constructions yield Raynaud orders as iterated extensions of rank \(p\) Hopf algebras; the exponential method obtains all Raynaud orders whose invariants satisfy a certain \(p\)-adic condition.

Readership

Advanced graduate students and research mathematicians working in formal groups, finite group schemes or local algebraic number theory and Galois module theory.

Table of Contents

• Introduction to polynomial formal groups and Hopf algebras
• Dimension one polynomial formal groups
• Dimension two polynomial formal groups and Hopf algebras
• Degree two formal groups and Hopf algebras
• \(p\)-Elementary group schemes--Constructions and Raynaud's theory