The generating series of Gromov-Witten invariants of a Calabi-Yau manifold are a priori defined as a formal series in the Novikov variable. The analytic properties of them (e.g., exhibited by Givental's I-functions at genus zero) tell that these formal series can actually be extended to complex-analytic objects living on a small neighborhood of the so-called large volume limit on the Kahler moduli space. Mirror symmetry   predicts that they are algebraic objects such as sections of certain vector bundles over the moduli of complex structure of the mirror Calabi-Yau manifold. 


 


I will describe  algebraic/geometric properties of these generating series, using examples such as elliptic curves and quintic threefolds. For the elliptic curve case we shall see that these series are (quasi-)modular forms which are fundamental objects in the study of tradictional number theory. The Gromov-Witten generating series of more general Calabi-Yau manifolds can be naturally regarded as generalizations of (quasi-)modular forms.