  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.
