Local systems and finite unitary and symplectic groups

For powers q of any odd prime p and any integer n≥2, we exhibit explicit local systems, on the affine line A¹ in characteristic p>0 if 2|n and on the affine plane A² if 2∤n, whose geometric monodromy groups are the finite symplectic groups Sp_2n(q). When n≥3 is odd, we show that the explicit rigid local systems on the affine line in characteristic p>0 constructed in [11] do have the special unitary groups SU_n(q) as their geometric monodromy groups as conjectured therein, and also prove another conjecture of [11] that predicted their arithmetic monodromy groups.