Special values of hypergeometric functions and periods of CM elliptic curves
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Abstract:
Let $X_0^6(1)/W_6$ be the Atkin–Lehner quotient of the Shimura curve $X_0^6(1)$ associated to a maximal order in an indefinite quaternion algebra of discriminant $6$ over $\mathbb {Q}$. By realizing modular forms on $X_0^6(1)/W_6$ in two ways, one in terms of hypergeometric functions and the other in terms of Borcherds forms, and using Schofer’s formula for values of Borcherds forms at CM-points, we obtain special values of certain hypergeometric functions in terms of periods of elliptic curves over $\overline {\mathbb {Q}}$ with complex multiplication.References
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Additional Information
- Yifan Yang
- Affiliation: Department of Mathematics, National Taiwan University and National Center for Theoretical Sciences, Taipei, Taiwan 106
- MR Author ID: 633505
- Email: yangyifan@ntu.edu.tw
- Received by editor(s): December 15, 2015
- Received by editor(s) in revised form: November 23, 2016
- Published electronically: December 28, 2017
- Additional Notes: The author was partially supported by Grant 106-2115-M-002-009-MY3 of the Ministry of Science and Technology, Taiwan (R.O.C.).
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 6433-6467
- MSC (2010): Primary 11F12; Secondary 11G15, 11G18, 33C05
- DOI: https://doi.org/10.1090/tran/7134
- MathSciNet review: 3814336