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

Kondo, Satoshi
Watari, Taizan
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.


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Citation Formats
S. Kondo and T. Watari, “String-Theory Realization of Modular Forms for Elliptic Curves with Complex Multiplication,” COMMUNICATIONS IN MATHEMATICAL PHYSICS, pp. 89–126, 2019, Accessed: 00, 2020. [Online]. Available: