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Mathematics of Computation

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A continued fraction of order twelve as a modular function


Authors: Yoonjin Lee and Yoon Kyung Park
Journal: Math. Comp. 87 (2018), 2011-2036
MSC (2010): Primary 11Y65, 11F03, 11R37, 11R04, 14H55
DOI: https://doi.org/10.1090/mcom/3259
Published electronically: September 28, 2017
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Abstract: We study a continued fraction $ U(\tau )$ of order twelve using the modular function theory. We obtain the modular equations of $ U(\tau )$ by computing the affine models of modular curves $ X(\Gamma )$ with $ \Gamma = \Gamma _1 (12) \cap \Gamma _0(12n)$ for any positive integer $ n$; this is a complete extension of the previous result of Mahadeva Naika et al. and Dharmendra et al. to every positive integer $ n$. We point out that we provide an explicit construction method for finding the modular equations of $ U(\tau )$. We also prove that these modular equations satisfy the Kronecker congruence relations. Furthermore, we show that we can construct the ray class field modulo $ 12$ over imaginary quadratic fields by using $ U(\tau )$ and the value $ U(\tau )$ at an imaginary quadratic argument is a unit. In addition, if $ U(\tau )$ is expressed in terms of radicals, then we can express $ U(r \tau )$ in terms of radicals for a positive rational number $ r$.


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Additional Information

Yoonjin Lee
Affiliation: Department of Mathematics, Ewha Womans University, Seoul 03760, South Korea
Email: yoonjinl@ewha.ac.kr

Yoon Kyung Park
Affiliation: Institute of Mathematical Sciences, Ewha Womans University, Seoul 03760, South Korea
Email: ykp@ewha.ac.kr

DOI: https://doi.org/10.1090/mcom/3259
Keywords: Ramanujan continued fraction, modular function
Received by editor(s): May 16, 2016
Received by editor(s) in revised form: January 11, 2017
Published electronically: September 28, 2017
Additional Notes: The first-named author is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2009-0093827) and also by the Korea government (MEST) (NRF-2017R1A2B2004574)
The second-named author was supported by RP-Grant 2016 of Ewha Womans University and Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2017R1D1A1B03029519)
Article copyright: © Copyright 2017 American Mathematical Society

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