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The Beilinson conjectures for CM elliptic curves via hypergeometric functions

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Abstract

We consider certain CM elliptic curves which are related to Fermat curves, and express the values of L-functions at \(s=2\) in terms of special values of generalized hypergeometric functions. We compare them and a similar result of Rogers–Zudilin with Otsubo’s regulator formulas, and give a new proof of the Beilinson conjectures originally due to Bloch.

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Acknowledgements

This paper is based on the author’s masters thesis at Chiba University. I am very grateful to Noriyuki Otsubo and Shigeki Matsuda for valuable advice. I would like to thank Mathew Rogers and Wadim Zudilin for their helpful comments.

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Authors and Affiliations

  1. Department of Mathematics and Informatics, Graduate School of Science, Chiba University, Yayoicho 1-33, Inage, Chiba, 263-8522, Japan

    Ryojun Ito

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  1. Ryojun Ito
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Correspondence to Ryojun Ito.

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Ito, R. The Beilinson conjectures for CM elliptic curves via hypergeometric functions. Ramanujan J 45, 433–449 (2018). https://doi.org/10.1007/s11139-016-9859-0

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  • DOI: https://doi.org/10.1007/s11139-016-9859-0

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