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A specific $2$-dimensional Galois representation of $G_{\mathbb{Q}_2}$ and its Langlands correspondence

I am interested in understanding a situation in (classical, not $p$-adic) local Langlands for $\mathrm{GL}_p(\mathbb{Q}_p)$. An example of it is as follows: Let $F=\mathbb{Q}_2$ and $E$ be the ...
Cheng-Chiang Tsai's user avatar