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Hypergeometric functions over $ \mathbb{F}_q$ and traces of Frobenius for elliptic curves


Authors: Rupam Barman and Gautam Kalita
Journal: Proc. Amer. Math. Soc. 141 (2013), 3403-3410
MSC (2010): Primary 11T24, 11G20
Published electronically: June 21, 2013
MathSciNet review: 3080163
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Abstract: We present explicit relations between the traces of Frobenius endomorphisms of certain families of elliptic curves and special values of $ {_{2}}F_1$-hypergeometric functions over $ \mathbb{F}_q$ for $ q \equiv 1 ($$ \text {mod}~6)$ and $ q \equiv 1 ($$ \text {mod}~4)$.


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Additional Information

Rupam Barman
Affiliation: Department of Mathematical Sciences, Tezpur University, Napaam-784028, Sonitpur, Assam, India
Email: rupamb@tezu.ernet.in

Gautam Kalita
Affiliation: Department of Mathematical Sciences, Tezpur University, Napaam-784028, Sonitpur, Assam, India
Email: gautamk@tezu.ernet.in

DOI: https://doi.org/10.1090/S0002-9939-2013-11617-5
Keywords: Gaussian hypergeometric series, elliptic curves, Frobenius endomorphisms
Received by editor(s): August 19, 2011
Received by editor(s) in revised form: December 21, 2011
Published electronically: June 21, 2013
Additional Notes: The first author thanks the Mathematical Institute, University of Heidelberg and Mathematics Center Heidelberg (MATCH), where the majority of this research was conducted. He is grateful to John H. Coates, R. Sujatha, Otmar Venjakob, and Anupam Saikia for their encouragement.
The second author was partially supported by an INSPIRE Fellowship of the Department of Science and Technology, Goverment of India.
Finally, the authors thank Ken Ono and the referee for helpful comments.
Communicated by: Ken Ono
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.