Breuil Modules for Raynaud Schemes

Let p be an odd prime, and let OK be the ring of integers in a finite extension K/Qp. Breuil has classified finite flat group schemes of type (p, . . . , p) over OK in terms of linear-algebraic objects that have come to be known as Breuil modules. This classification can be extended to the case of finite flat vector space schemes G over OK . When G has rank one, the generic fiber of G corresponds to a Galois character, and we explicitly determine this character in terms of the Breuil module of G. Special attention is paid to Breuil modules with descent data corresponding to characters of Gal(Qp/Qpd ) that become finite flat over a totally ramified extension of degree p − 1; these arise in Gee’s work on the weight in Serre’s conjecture over totally real fields.