Hello,

Nowadays, I think we have some classification of integral structure in semistable representation via Liu's $(\varphi, \hat{G})$-modules or via Caruso's $(\varphi, \tau)$-modules. I must say that because of lack of time and motivation, I didn't read their papers, nor the ones by Breuil or Kisin, so I know almost nothing about integral p-adic Hodge theory.

So my question is the following :

given a lattice $T$ in a semistable representation, is there a way to read the Hodge-Tate weights on the corresponding object (namely the associated $(\varphi, \hat{G})$-module) or the $(\varphi, \tau)$-module) ?

integralp-adic Hodge theory. In any case, the answer is yes, because you can recover the weakly admissible Fontaine module associated with the semi-stable representation from Liu's $(\varphi,\hat{G})$-module (and probably Caruso's as well). In fact, you only need the $\varphi$ part of it, if I remember correctly. – Keerthi Madapusi Pera Nov 11 '11 at 19:33