Patching and p-adic local Langlands

The p-adic local Langlands correspondence is well understood for GL_2(Q_p), but appears much more complicated when considering GL_n(F), where either n>2 or F is a finite extension of 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 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.

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