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$\text{mod}~p$ non-abelian Lubin–Tate theory for 
$\text{GL}_{2}(\mathbb{Q}_{p})$
We analyse the 
$\text{mod}~p$ étale cohomology of the Lubin–Tate tower both with compact support and without support. We prove that there are no supersingular representations in the 
$H_{c}^{1}$ of the Lubin–Tate tower. On the other hand, we show that in 
$H^{1}$ of the Lubin–Tate tower appears the 
$\text{mod}~p$ local Langlands correspondence and the 
$\text{mod}~p$ 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.
