  We study examples of fourth-order Picard-Fuchs operators that are Hadamard
products of two second-order Picard-Fuchs operators. Each second-order
Picard-Fuchs operator is associated with a family of elliptic curves, and the
Hadamard product computes period integrals on the fibred product of the two
elliptic surfaces. We construct 3-cycles on this geometry as the union of
2-cycles in the fibre over contours on the base. We then use the special
Lagrangian condition to constrain the contours on the base. This leads to a
construction that is reminiscent of spectral networks and exponential networks
that have previously appeared in string theory literature.
