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

TL;DR

This paper explores the connection between string theory and modular forms associated with elliptic curves having complex multiplication, revealing a string-theoretic interpretation of these mathematical objects.

Contribution

It demonstrates that certain modular forms for CM elliptic curves can be realized through string theory models using diagonal N=(2,2) superconformal field theories.

Findings

Modular forms correspond to Boltzmann-weighted sums of U(1) charges in string theory.

String theory provides a physical realization of modular forms for CM elliptic curves.

The approach links number theory and string theory via conformal field theory computations.

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.