Crystalline Chebotarëv density theorems

Abstract: We formulate a conjectural analogue of Chebotar\"ev's density theorem for convergent $F$-isocrystals over a smooth geometrically irreducible curve defined over a finite field using the Tannakian formalism. We prove this analogue for several large classes, including direct sums of isoclinic convergent $F$-isocrystals and semi-simple convergent $F$-isocrystals which have an overconvergent extension and such that the semi-simplification of their pull-back to a sufficient small non-empty open sub-curve has abelian monodromy. In order to prove the latter, we also prove an overconvergent analogue of Chebotar\"ev's density theorem for semi-simple overconvergent $F$-isocrystals.