Compositio Mathematica

Research Article

On lattices in semi-stable representations: a proof of a conjecture of Breuil

Tong Liua1

a1 Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104, USA (email:


For p≥3 an odd prime and a nonnegative integer rp−2, we prove a conjecture of Breuil on lattices in semi-stable representations, that is, the anti-equivalence of categories between the category of strongly divisible lattices of weight r and the category of Galois stable $\mathbb {Z}_p$-lattices in semi-stable p-adic Galois representations with Hodge–Tate weights in {0,…,r}.

(Received June 26 2006)

(Accepted May 21 2007)

(Online publication January 23 2008)


  • p-adic representations;
  • semi-stable;
  • strongly divisible lattices

2000 Mathematics subject classification

  • 14F30;
  • 14L05 (primary)