Let G be the group of rational points of a p-adic reductive group and H its associated pro-p Iwahori-Hecke algebra over a field k of characteristic p. The mod-p Langlands program aims to relate the representation theory of G over k to that of the absolute Galois group of Q_p. The representations of G in this context are however still very poorly understood. On the other hand, the H-modules are much better understood and there even are results relating them to Galois representations. In earlier work, we investigated the so-called Gorenstein projective model structure on the category of H-modules and its associated homotopy category Ho(H). Assuming G has semisimple rank 1, we will explain in this talk how this category Ho(H) identifies with the singularity category of a suitable scheme parametrising Galois representations, building on earlier work of Pépin-Schmidt. After taking a suitable notion of support, this recovers (most of) the mod-p Langlands correspondence for GL_2(Q_p).
