  We carry out the SYZ program for the local Calabi--Yau manifolds of type
$\widetilde{A}$ by developing an equivariant SYZ theory for the toric
Calabi--Yau manifolds of infinite-type. Mirror geometry is shown to be
expressed in terms of the Riemann theta functions and generating functions of
open Gromov--Witten invariants, whose modular properties are found and studied
in this article. Our work also provides a mathematical justification for a
mirror symmetry assertion of the physicists Hollowood--Iqbal--Vafa.
