Generalized Hodge-Tate weights of an arbitrary p-adic Galois representation

Question

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!

Answer 1

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".