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 A(1) in characteristic p > 0 if 2 vertical bar n and on the affine plane A(2) if 2 vertical bar 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 SUn (q) as their geometric monodromy groups as conjectured therein, and also prove another conjecture of [11] that predicted their arithmetic monodromy groups. (C) 2019 Elsevier Inc. All rights reserved.