Understand the $p$-adic local Langlands correspondence with examples

Question

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?

Answer 1

Let's look at the case of representations associated to modular forms. I'm going to switch the roles of $\ell$ and $p$, because I find $\ell$-adic Hodge theory disturbing; so I'm going to look at $\rho_{f, \ell} |_{G_{\mathbb{Q}_p}}$.

For $\ell \ne p$, we can attach a Weil--Deligne representation to $\rho_{f, \ell} |_{G_{\mathbb{Q}_p}}$ via Grothendieck abstract monodromy. This WD representation has coefficients in $\overline{\mathbb{Q}}_\ell$, but the definition of a WD representation doesn't depend on the topology on the coefficient field, so we can transport it along a field isomorphism $\overline{\mathbb{Q}}_\ell \cong \mathbb{C}$ to get a complex-valued Weil--Deligne representation. The local-global compatibility theorem (due to Carayol in this case) tells us that this WD representation is the one associated by local Langlands to the smooth complex $GL_2(\mathbb{Q}_p)$-representation $\pi_{f, p}$ (the local factor of $f$ at $p$).

Now, there are multiple ways of extending this to cover $\ell = p$. One approach is the following: via Fontaine's functor $D_{\mathrm{pst}}$, we can attach a Weil--Deligne representation to $\rho_{f, p}|_{G_{\mathbb{Q}_p}}$; and it is known (by a theorem of Saito, IIRC) that this Weil--Deligne representation (again, transported via a field isomorphism $\overline{\mathbb{Q}}_p \cong \mathbb{C}$) is the one associated to $\pi_p$ by local Langlands. So that's a meaningful statement of "local-global compatibility for $\ell = p$" which doesn't involve p-adic Banach spaces.

However, this formulation isn't quite the whole story, because:

• Fontaine's $D_{\mathrm{pst}}$ functor only applies to a subclass of $p$-adic representations of $G_{\mathbb{Q}_p}$ (the de Rham ones); this includes all the ones from modular forms, but it misses lots of other interesting objects (e.g. representations associated to non-classical overconvergent eigenforms).
• Even when $V$ is de Rham, the Weil--Deligne representation associated to $V$ doesn't uniquely determine $V$ up to isomorphism, because it forgets the Hodge filtration, and in some cases there are multiple non-isomorphic choices of filtrations for a given $V$. (This is precisely what the $\mathcal{L}$-invariant parametrises.)

This motivates the formulation of $p$-adic Langlands in terms of Banach-space representations, which are "rich enough" to match up with the whole category of p-adic representations of $G_{\mathbb{Q}_p}$. Some Banach-space representations are completions of smooth (or locally-algebraic) representations with respect to an invariant norm, which gives the link with the classical formulation of Langlands; but many Banach-space representations aren't of this type, and more subtly, in some cases the same smooth representation can admit multiple invariant norms (giving different Banach representations as the completions), which is, again, parametrised by the $\mathcal{L}$-invariant.