I would like to understand in depth why in the $l=p$ case of the local Langlands correspondence must consider $p$-adic representations instead of a complex representation. As far as I know, the $p$-adic representations
 1. retrieve information that loses complex matches (like $\mathcal{L}$ invariant),
 2. also allows for local-to-global correspondence.

Can someone give examples of 1. and 2. in $Gl_{1}$ and $Gl_{2}$ case?