  We study the cohomology with modular coefficients of Deligne-Lusztig
varieties associated to Coxeter elements. Under some torsion-free assumption on
the cohomology we derive several results on the principal l-block of a finite
reductive group G(F_q) when the order of q modulo l is assumed to be the
Coxeter number. These results include the determination of the planar embedded
Brauer tree of the block (as conjectured by Hiss, L\"ubeck and Malle) and the
derived equivalence predicted by the geometric version of Brou\'e's conjecture.
