String-Theory Realization of Modular Forms for Elliptic Curves with Complex Multiplication

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Kondo S., Watari T.

COMMUNICATIONS IN MATHEMATICAL PHYSICS, vol.367, no.1, pp.89-126, 2019 (SCI-Expanded) identifier identifier


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 (qL0-c/24-weighted) sum of U(1) charges with FeiF insertion computed in the Ramond sector.