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arXiv · 1801.07464

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

Satoshi Kondo·Taizan Watari
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Abstract

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.

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Satoshi Kondo, Taizan Watari. 2018-01-23. String-theory Realization of Modular Forms for Elliptic Curves with Complex Multiplication. https://doi.org/10.1007/s00220-019-03302-0

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