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

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!

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".