Rigid connections on P1 via the Bruhat–Tits building

We apply the theory of fundamental strata of Bremer and Sage to find cohomologically rigid (Formula presented.) -connections on the projective line, generalising the work of Frenkel and Gross. In this theory, one studies the leading term of a formal connection with respect to the Moy–Prasad filtration associated to a point in the Bruhat–Tits building. If the leading term is regular semisimple with centraliser a (not necessarily split) maximal torus (Formula presented.), then we have an (Formula presented.) -toral connection. In this language, the irregular singularity of the Frenkel–Gross connection gives rise to the homogeneous toral connection of minimal slope associated to the Coxeter torus (Formula presented.). In the present paper, we consider connections on (Formula presented.) which have an irregular homogeneous (Formula presented.) -toral singularity at zero of slope (Formula presented.), where (Formula presented.) is the Coxeter number and (Formula presented.) is a positive integer coprime to (Formula presented.), and a regular singularity at infinity with unipotent monodromy. Our main result is the characterisation of all such connections which are rigid.