  In this thesis we study asymptotic behavior of projective embeddings of
abelian varieties and their amoebas. The projective embeddings are given by
theta functions. It is known that a Lagrangian fibration of the abelian variety
determines a basis of theta functions. After reviewing the relation from the
viewpoint of geometric quantization and mirror symmetry, we prove that the
Lagrangian fibration of the abelian variety can be approximated by the moment
maps of natural torus action in a suitable sense. In the final section we
discuss a part of the result in more general setting.
