# algebraic number theory – A Tate-Sen theorem mod \$p\$

Let $$p$$ be a prime number, $$G=textrm{Gal}(overline{mathbb{Q}}_p/mathbb{Q}_p)$$, and $$chi:Grightarrowmathbb{Z}_p^times$$ the cyclotomic character. Let $$mathbb{C}_p$$ denote the completion of the algebraic closure of $$mathbb{Q}_p$$ and $$mathcal{O}_{mathbb{C}_p}$$ denote its ring of integers.

The Tate-Sen theorem implies that (among many other things) $$H^0(G,mathbb{C}_p(chi))=0$$.

Question: Is $$H^0(G,mathcal{O}_{mathbb{C}_p}(chi)otimes mathbb{F}_p)=0$$?

If the answer is no, do the Galois invariants generate (over $$mathcal{O}_{mathbb{C}_p}$$), or are they killed by some power of $$p$$ less than 1?

Motivation: Basically, I am trying to understand if there is some remannt of Hodge-Tate theory modulo $$p$$.

Thanks!