  For an elliptic curve E over an abelian extension k/K with CM by K of Shimura
type, the L-functions of its [k:K] Galois representations are Mellin transforms
of Hecke theta functions; a modular parametrization (surjective map) from a
modular curve to E pulls back the 1-forms on E to give the Hecke theta
functions. This article refines the study of our earlier work and shows that
certain class of chiral correlation functions in Type II string theory with
[E]_C (E as real analytic manifold) as a target space yield the same Hecke
theta functions as objects on the modular curve. The Kahler parameter of the
target space [E]_C in string theory plays the role of the index (partially
ordered) set in defining the projective/direct limit of modular curves.
