Modular parametrization as Polyakov path integral: cases with CM elliptic curves as target spaces

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.