Communications in Mathematical Physics

, Volume 367, Issue 1, pp 89–126 | Cite as

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

  • Satoshi Kondo
  • Taizan WatariEmail author


It is known that the L-function of an elliptic curve defined over \({\mathbb{Q}}\) 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 \({\mathcal{N}=(2,2)}\) 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 \({Fe^{\pi i F}}\) insertion computed in the Ramond sector.


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We are grateful to T. Abe and K. Hori for discussion and useful comments. This work is supported by WPI Initiative, MEXT, Japan (SK, TW).


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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Middle East Technical UniversityKalkanliTurkey
  2. 2.Kavli Institute for the Physics and Mathematics of the UniverseUniversity of TokyoKashiwaJapan

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