Geometric approach to the local Jacquet-Langlands correspondence

In this paper, we give a purely geometric approach to the local Jacquet-Langlands correspondence for ${\rm GL}(n)$ over a $p$-adic field, under the assumption that the invariant of the division algebra is $1/n$. We use the $\ell$-adic \'etale cohomology of the Drinfeld tower to construct the correspondence at the level of the Grothendieck groups with rational coefficients. Moreover, assuming that $n$ is prime, we prove that this correspondence preserves irreducible representations. This gives a purely local proof of the local Jacquet-Langlands correspondence in this case. We need neither a global automorphic technique nor detailed classification of supercuspidal representations of ${\rm GL}(n)$.