  In this expository note we discuss some arithmetic aspects of the mirror
symmetry for plane cubic curves. We also explain how the Picard-Fuchs equation
can be used to reveal part of these arithmetic properties. The application of
Picard-Fuchs equations in studying the genus zero Gromov-Witten invariants of
more general Calabi-Yau varieties and the Weil-Petersson geometry on their
moduli spaces will also be discussed.
