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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) ?

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    $\begingroup$ The Hodge-Tate weights make sense already without the integral structure, so it's unclear to me why you need integral p-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. $\endgroup$
    – Keerthi Madapusi
    Commented Nov 11, 2011 at 19:33
  • $\begingroup$ I am mostly interested in torsion representations, that's why I only want to consider the integral structure. $\endgroup$
    – user16131
    Commented Nov 11, 2011 at 19:41

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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).

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  • $\begingroup$ Sorry for taking so long to comment : is there a proof of this fact written somewhere ? $\endgroup$
    – user16131
    Commented Jun 6, 2012 at 14:48
  • $\begingroup$ The point is that $\varphi^*\mathfrak{M}/E(u)\varphi^*\mathfrak{M}\left[\frac{1}{p}\right]$ is canonically identified with the $(\varphi,N)$-module over $K$ attached to the Galois representation. Under this identification, the Hodge filtration has the description given above. See Prop. 1.2.8 in Kisin's paper 'Crystalline representations and F-crystals'. $\endgroup$
    – Keerthi Madapusi
    Commented Jun 18, 2012 at 1:04

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