  We prove Rietsch's mirror conjecture that the Dubrovin quantum connection for
minuscule flag varieties is isomorphic to the character D-module of the
Berenstein-Kazhdan geometric crystal. The idea is to recognize the quantum
connection as Galois and the geometric crystal as automorphic. We reveal
surprising relations with the works of Frenkel-Gross, Heinloth-Ng\^o-Yun, and
Zhu on Kloosterman sheaves. The isomorphism comes from global rigidity results
where Hecke eigensheaves are determined by their local ramification. As
corollaries we obtain combinatorial identities for counts of rational curves
and the Peterson variety presentation of the small quantum cohomology ring.
