  It is known that the L-function of an elliptic curve defined over Q is given
by the Mellin transform of a modular form of weight 2. Does that modular form
have anything to do with string theory? In this article, we address a question
along this line for elliptic curves that have complex multiplication defined
over number fields. So long as we use diagonal rational N=(2,2) superconformal
field theories for the string-theory realizations of the elliptic curves, the
weight-2 modular form turns out to be the Boltzmann-weighted
(q^{L_0-c/24}-weighted) sum of U(1) charges with F e^{ \pi i F} insertion
computed in the Ramond sector.
