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$p$
-adic Langlands program for
$\operatorname{GL}_{2}(\mathbb{Q}_{p})$
We present a new construction of the
$p$
-adic local Langlands correspondence for
$\operatorname{GL}_{2}(\mathbb{Q}_{p})$
via the patching method of Taylor–Wiles and Kisin. This construction sheds light on the relationship between the various other approaches to both the local and the global aspects of the
$p$
-adic Langlands program; in particular, it gives a new proof of many cases of the second author’s local–global compatibility theorem and relaxes a hypothesis on the local mod
$p$
representation in that theorem.
