On mod p non-abelian Lubin-Tate theory for GL_2(Q_p)

TL;DR

This paper investigates the mod p étale cohomology of the Lubin-Tate tower for GL_2(Q_p), revealing the absence of supersingular representations in certain cohomology groups and establishing connections with mod p local Langlands and Jacquet-Langlands correspondences.

Contribution

It demonstrates the non-existence of supersingular representations in H^1_c and links H^1 to key mod p local correspondences, advancing understanding of mod p non-abelian Lubin-Tate theory.

Findings

No supersingular representations in H^1_c of the Lubin-Tate tower.

H^1 of the Lubin-Tate tower contains the mod p local Langlands correspondence.

H^1 also exhibits the mod p local Jacquet-Langlands correspondence.

Abstract

We analyse the mod p \'etale cohomology of the Lubin-Tate tower both with compact support and without support. We prove that there are no supersingular representations in the H^1_c of the Lubin-Tate tower. On the other hand, we show that in H^1 of the Lubin-Tate tower appears the mod p local Langlands correspondence and the 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.