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?

Let $\rho:G_{\mathbb{Q}}\rightarrow \mathbf{Gl}_{n}(\mathbb{Q}_{p})$. I would like to understand in depth why the local Langlands correspondence for $\rho_{|\mathbb{Q}_{p}}$ 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 representations (like $\mathcal{L}$ invariant),
2. also allows for local-global correspondence.

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

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 correspondance.

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

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?

I would like to understand in depth why in the $l=p$ case of the locale 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 correspondance.

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

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 correspondance.

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