Let $V$ be a continuous representation of the absolute Galois group of $\mathbb{Q}\_p$ with coefficients in $\mathbb{Q}\_p$. The theory of Sen attaches to $V$ generalized Hodge-Tate weights which are elements in $\overline{\mathbb{Q}}\_p$.


**My question is**: are the generalized Hodge-Tate weight always elements of $\overline{\mathbb{Z}}\_p$?


In his infinite fern paper from 1997, Mazur says he does not know of an example (at least in dimension 2) where the weights are not in $\overline{\mathbb{Z}}\_p$. Personally, I do not recall of ever seeing an example with weights not lying in $\mathbb{Z}\_p$. 


Thanks!



Generalized Hodge-Tate weights of an arbitrary p-adic Galois representation
If $K$ is a finite extension of $Q\_p$, let $G\_K = Gal(Q\_p^{alg}/K)$. Let $\chi: G\_K \to Z\_p^\times$ be the cyclotomic character. If $n \geq 1$ and $K$ is large enough, then $\chi(G\_K) \subset 1+p^{n+1} Z\_p$ so that $\chi(g)^{1/p^n}$ (using the binomial series) converges and makes sense if $g \in G\_K$. This gives you a character of $G\_K$ with HT weight $1/p^n$. Now you can induce it to $G\_{Qp}$ to get a representation with HT weights $1/p^n$.


For a more general construction, see lemma 2.1.3 of G. Di Matteo's "On admissible tensor products in p-adic Hodge theory".


For a result in the opposite direction, see remark 4.1.3 of L. Berger & P. Colmez "Familles de représentations de de Rham et monodromie p-adique".


