A continued fraction of order twelve as a modular function
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- by Yoonjin Lee and Yoon Kyung Park PDF
- Math. Comp. 87 (2018), 2011-2036 Request permission
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 $r$.References
- Bryden Cais and Brian Conrad, Modular curves and Ramanujan’s continued fraction, J. Reine Angew. Math. 597 (2006), 27–104. MR 2264315, DOI 10.1515/CRELLE.2006.063
- Bumkyu Cho and Ja Kyung Koo, Construction of class fields over imaginary quadratic fields and applications, Q. J. Math. 61 (2010), no. 2, 199–216. MR 2646085, DOI 10.1093/qmath/han035
- B. N. Dharmendra, M. R. Rajesh Kanna and R. Jagadeesh, On continued fraction of order twelve, Pure Math. Sci. 1 (2012), 197–205.
- Bumkyu Cho, Ja Kyung Koo, and Yoon Kyung Park, Arithmetic of the Ramanujan-Göllnitz-Gordon continued fraction, J. Number Theory 129 (2009), no. 4, 922–947. MR 2499414, DOI 10.1016/j.jnt.2008.09.018
- Alice Gee and Mascha Honsbeek, Singular values of the Rogers-Ramanujan continued fraction, Ramanujan J. 11 (2006), no. 3, 267–284. MR 2249499, DOI 10.1007/s11139-006-8477-7
- Nobuhiko Ishida and Noburo Ishii, The equations for modular function fields of principal congruence subgroups of prime level, Manuscripta Math. 90 (1996), no. 3, 271–285. MR 1397657, DOI 10.1007/BF02568306
- Chang Heon Kim and Ja Kyung Koo, Generation of Hauptmoduln of $\Gamma _1(N)$ by Weierstrass units and application to class fields, Cent. Eur. J. Math. 9 (2011), no. 6, 1389–1402. MR 2836730, DOI 10.2478/s11533-011-0080-5
- Daniel S. Kubert and Serge Lang, Modular units, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 244, Springer-Verlag, New York-Berlin, 1981. MR 648603, DOI 10.1007/978-1-4757-1741-9
- Yoonjin Lee and Yoon Kyung Park, Modularity of a Ramanujan-Selberg continued fraction, J. Math. Anal. Appl. 438 (2016), no. 1, 373–394. MR 3462583, DOI 10.1016/j.jmaa.2016.01.065
- M. S. Mahadeva Naika, B. N. Dharmendra, and K. Shivashankara, A continued fraction of order twelve, Cent. Eur. J. Math. 6 (2008), no. 3, 393–404. MR 2425001, DOI 10.2478/s11533-008-0031-y
- M. S. Mahadeva Naika, S. Chandankumar, and K. Sushan Bairy, Some new identities for a continued fraction of order 12, South East Asian J. Math. Math. Sci. 10 (2012), no. 2, 129–140. MR 2978181
- M. S. Mahadeva Naika, S. Chandankumar and K. Sushan Bairy, New identites for ratios of Ramanujan’s theta function (communicated).
- Ken Ono, The web of modularity: arithmetic of the coefficients of modular forms and $q$-series, CBMS Regional Conference Series in Mathematics, vol. 102, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2004. MR 2020489
- Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Kanô Memorial Lectures, No. 1, Iwanami Shoten Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. Publications of the Mathematical Society of Japan, No. 11. MR 0314766
Additional Information
- Yoonjin Lee
- Affiliation: Department of Mathematics, Ewha Womans University, Seoul 03760, South Korea
- MR Author ID: 689346
- ORCID: 0000-0001-9510-3691
- Email: yoonjinl@ewha.ac.kr
- Yoon Kyung Park
- Affiliation: Institute of Mathematical Sciences, Ewha Womans University, Seoul 03760, South Korea
- MR Author ID: 836403
- Email: ykp@ewha.ac.kr
- 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) - © Copyright 2017 American Mathematical Society
- Journal: Math. Comp. 87 (2018), 2011-2036
- MSC (2010): Primary 11Y65, 11F03, 11R37, 11R04, 14H55
- DOI: https://doi.org/10.1090/mcom/3259
- MathSciNet review: 3787400