Author: Eiko

Tags: p-adic, hodge theory, representation

Time: 2024-11-21 20:58:49 - 2024-11-21 20:58:49 (UTC)

Let $K$ be a field, $\bar{K}/K$ the separable closure and $G_K$ the absolute Galois group of $K$, $l,p$ are fixed primes.

Definition An $l$-adic representation of $G_K$ is a finite dimensional ${\mathbb{Q}}_l$-vector space $V$ with a continuous action of $G_K$, $\rho :G_K\to \mathrm{GL}(V)\cong {\mathrm{GL}}_d({\mathbb{Q}}_l)$, $d=\dim V$.

Since $G_K$ is compact, we can always choose an ${\mathbb{Z}}_l$-lattice $T\subset V$ that is $G_K$-stable, i.e. $\rho (G_K)T\subset T$. Then we can define the $l$-adic Galois representation $V_{{\mathbb{Q}}_l}:=T{\otimes }_{{\mathbb{Z}}_l}{\mathbb{Q}}_l$.