Joint IAS/Princeton University Number Theory Seminar

The \(p\)-adic local Langlands correspondence is well understood for \(\mathrm{GL}_2(\mathbb Q_p)\), but appears much more complicated when considering \(\mathrm{GL}_n(F)\), where either \(n>2\) or \(F\) is a finite extension of \(\mathbb Q_p\). I will discuss joint work with Matthew Emerton, Toby Gee, David Geraghty, Vytautas Paskunas and Sug Woo Shin, in which we approach the p-adic local Langlands correspondence for \(\mathrm{GL}_n(F)\) using global methods. The key ingredient is Taylor-Wiles-Kisin patching of completed cohomology. This allows us to prove many new cases of the Breuil-Schneider conjecture.