Potential semi-stability of p-adic representations coming from algebraic geometry
Last week, I participated in a summer school on arithmetic geometry and Langlands program. Although this is not my field of interests, I always give some respect those who work in those fields. I also did gave a motivational TA talk about couting points on elliptic curves. To be honest, I do not expect to learn much from this school but I got enough inspiration to read arithmetic stuff, which I am going to write. Since I am an algebraic geometer, I'd like to present things in a geometric way. Let $p$ be a prime number and denote $\mathbb{Q}_p$ the $p$-adic local field. Let $K$ be a finite extension of $\mathbb{Q}_p$. Let $G_K = \mathrm{Gal}(\overline{K}/K)$ the absolute Galois group. A $\ell$- adic representation of $G_K$ is a continuous morphism (of groups) $$G_K \longrightarrow \mathrm{GL}_n(V),$$ where $V$ is a finite-dimensional $\mathbb{Q}_{\ell}$-vector space with $\ell \neq p$ a prime. We denote by $K^{unr}$ the maximal unramified extension and $I_K = \mathrm{Gal}(\overline