Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
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ON CONTINUED FRACTIONS, FUNDAMENTAL UNITS AND CLASS NUMBERS OF REAL QUADRATIC FUNCTION FIELDS

  • Received : 2013.12.11
  • Accepted : 2014.04.16
  • Published : 2014.05.15
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Abstract

We examine fundamental units of quadratic function fields from continued fraction of $\sqrt{D}$. As a consequence, we give another proof of geometric analog of Ankeny-Artin-Chowla-Mordell conjecture and bounds for class number, and study real quadratic function fields of minimal type with quasi-period 4.

Keywords

Acknowledgement

Supported by : Chungnam National University

References

  1. S. Bae, Real quadratic function fields of Richaud-Degert type with ideal class number one, Proc. AMS 140 (2012), 403-414. https://doi.org/10.1090/S0002-9939-2011-10910-9
  2. D. Byeon and S. Lee, A note on units of real quadratic fields, Bull. Korean Math. Soc. 49 (2012), 767-774. https://doi.org/10.4134/BKMS.2012.49.4.767
  3. C. Friesen, Continued fractions and real quadratic function fields, Doctoral Thesis, Brown University (1989).
  4. R. Hashimoto, Ankeny-Artin-Chowla conjecture and continued fraction, J. Number Th. 90 (2001), 143-153. https://doi.org/10.1006/jnth.2001.2652
  5. C. N. Hsu, Estimates for coefficients of L-functions for function fields, Finite Fields and their Appl. 5 (1999), 76-88. https://doi.org/10.1006/ffta.1998.0234
  6. F. Kawamoto and K. Tomita, Continued fractions and certain real quadratic fields of minimal type, J. Math. Soc. Japan 60 (2008), 865-903. https://doi.org/10.2969/jmsj/06030865
  7. R. A. Mollin and H. C. Williams, A complete generalization of Yokoi's p-invariants, Colloquium Mathematicum 63 (1992), 285-294.
  8. G. D. Villa Salvador, Topics in the theory of algebraic function fields, Birkhauser (2006).
  9. L. Washington, Introduction to cyclotomic fields GTM vol 83 Springer-Verlag, 1982.
  10. H. Yokoi, The fundamental unit and bounds for class numbers of real quadratic fields, Nagoya Math. J. 124 (1991), 181-197. https://doi.org/10.1017/S0027763000003834
  11. H. Yokoi, New invariants and class number problem in real quadratic fields, Nagoya Math. J. 132 (1993), 175-197. https://doi.org/10.1017/S0027763000004700
  12. J. Yu and J. K. Yu, A note on a geometric analogue of Ankeny-Artin-Chowla's conjecture, Contemporary Math. A. M. S. 210 (1998), 101-105. https://doi.org/10.1090/conm/210/02787
  13. R. Zuccherato, The continued fraction algorithm and regulator for quadratic function fields of characteristic 2, J. Algebra 190 (1997), 563-587. https://doi.org/10.1006/jabr.1996.6985