String-theory Realization of Modular Forms for Elliptic Curves with Complex Multiplication
High Energy Physics - Theory
2019-03-27 v2 Number Theory
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.
Cite
@article{arxiv.1801.07464,
title = {String-theory Realization of Modular Forms for Elliptic Curves with Complex Multiplication},
author = {Satoshi Kondo and Taizan Watari},
journal= {arXiv preprint arXiv:1801.07464},
year = {2019}
}
Comments
48 pages; minor corrections and improvements in v2