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We analyse the
étale cohomology of the Lubin–Tate tower both with compact support and without support. We prove that there are no supersingular representations in the
of the Lubin–Tate tower. On the other hand, we show that in
of the Lubin–Tate tower appears the
local Langlands correspondence and the
local Jacquet–Langlands correspondence, which we define in the text. We discuss the local-global compatibility part of the Buzzard–Diamond–Jarvis conjecture which appears naturally in this context.
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