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) ? 



Integral p-adic Hodge theory
Yes, there is.


Actually, the Hodge-Tate weights can be read only from $\varphi$ as follows. Let $\mathfrak M$ be the corresponding $(\varphi,\hat G)$-module or $(\varphi,\tau)$-modul. Denote by $\varphi(\mathfrak M)$ the $\mathfrak S$-module generated by the image of $\varphi$. Then, since by definition $E(u)^r \mathfrak M \subset \varphi(\mathfrak M)$ (here $E(u)$ is the minimal polynomial of the fixed uniformizer $\pi$ of $K$ and $r$ is some nonnegative integer), the quotient $\mathfrak M[1/p]/\varphi(\mathfrak M)[1/p]$ is a direct sum of pieces of the form $\mathfrak S[1/p]/E(u)^{n\_i}\mathfrak S[1/p]$ for some integers $n\_i$. These integers are precisely the Hodge-Tate weights of the representation (or the opposite of them depending on your convention).


