  We use hypergeometric sheaves on $G_m/F_q$, which are particular sorts of
rigid local systems, to construct explicit local systems whose arithmetic and
geometric monodromy groups are the finite general linear groups $GL_n(q)$ for
any $n \ge 2$ and and any prime power $q$, so long as $q > 3$ when $n=2$. This
paper continues a program of finding simple (in the sense of simple to
remember) families of exponential sums whose monodromy groups are certain
finite groups of Lie type, cf. [Gr], [KT1], [KT2], [KT3] for (certain) finite
symplectic and unitary groups, or certain sporadic groups, cf. [KRL], [KRLT1],
[KRLT2], [KRLT3]. The novelty of this paper is obtaining $GL_n(q)$ in this
hypergeometric way. A pullback construction then yields local systems on
$A^1/F_q$ whose geometric monodromy groups are $SL_n(q)$. These turn out to
recover a construction of Abhyankar.
