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Periodic points of algebraic functions and Deuring’s class number formula

  • Patrick MortonEmail author
Article

Abstract

The exact set of periodic points in \(\overline{\mathbb {Q}}\) of the algebraic function \({\hat{F}}(z)=(-1\pm \sqrt{1-z^4})/z^2\) is shown to consist of the coordinates of certain solutions \((x,y)=(\pi , \xi )\) of the Fermat equation \(x^4+y^4=1\) in ring class fields \(\Omega _f\) over imaginary quadratic fields \(K=\mathbb {Q}(\sqrt{-d})\) of odd conductor f, where \(-d =d_K f^2 \equiv 1\) (mod 8). This is shown to result from the fact that the 2-adic function \(F(z)=(-1+ \sqrt{1-z^4})/z^2\) is a lift of the Frobenius automorphism on the coordinates \(\pi \) for which \(|\pi |_2<1\), for any \(d \equiv 7\) (mod 8), when considered as elements of the maximal unramified extension \(\textsf {K}_2\) of the 2-adic field \(\mathbb {Q}_2\). This gives an interpretation of the case \(p=2\) of a class number formula of Deuring. An algebraic method of computing these periodic points and the corresponding class equations \(H_{-d}(x)\) is given that is applicable for small periods. The pre-periodic points of \({\hat{F}}(z)\) in \(\overline{\mathbb {Q}}\) are also determined.

Keywords

Periodic points Algebraic function Class number formula Modular function Ring class fields 

Mathematics Subject Classification

11D41 11G07 11G15 14H05 

Notes

References

  1. 1.
    Brillhart, J., Morton, P.: Class numbers of quadratic fields, Hasse invariants of elliptic curves, and the supersingular polynomial. J. Number Theory 106, 79–111 (2004)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Chandrasekharan, K.: Elliptic Functions, Grundlehren der math. Wissenschaften 281. Springer, Berlin (1985)Google Scholar
  3. 3.
    Cox, David A.: Primes of the Form \(x^2+ny^2\); Fermat, Class Field Theory, and Complex Multiplication. Wiley, New York (1989)Google Scholar
  4. 4.
    Deuring, M.: Die Typen der Multiplikatorenringe elliptischer Funktionenkörper. Abh. Math. Sem. Univ. Hamburg 14(1), 197–272 (1941)CrossRefGoogle Scholar
  5. 5.
    Deuring, M.: Die Anzahl der Typen von Maximalordnungen einer definiten Quaternionenalgebra mit primer Grundzahl. Jahresbericht der Deutschen Mathematiker-Vereinigung 54, 24–41 (1944)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Deuring, M.: Teilbarkeitseigenschaften der singulären Moduln der elliptischen Funktionen und die Diskriminante der Klassengleichung. Comment. Math. Helv. 19, 74–82 (1946)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gross, B.H., Zagier, D.B.: On singular moduli. J. Reine Angew. Math. 355, 191–220 (1985)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Kani, E.: Idoneal numbers and some generalizations. Ann. Sci. Math. Québec 35(2), 197–227 (2011)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Lauter, K., Viray, B.: On singular moduli for arbitrary discriminants. Int. Math. Res. Notices IMRN 19, 9206–9250 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Lynch, R., Morton, P.: The quartic Fermat equation in Hilbert class fields of imaginary quadratic fields. Int. J. Number Theory 11, 1961–2017 (2015)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Morton, P.: Solutions of the cubic Fermat equation in ring class fields of imaginary quadratic fields (as periodic points of a \(3\)-adic algebraic function). Int. J. Number Theory 12, 853–902 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Morton, P.: Solutions of diophantine equations as periodic points of \(p\)-adic algebraic functions, I. New York J. Math 22, 715–740 (2016)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Schertz, R.: Complex Multiplication, New Mathematical Monographs, vol. 15. Cambridge University Press, Cambridge (2010)CrossRefGoogle Scholar
  14. 14.
    Schoeneberg, B.: Elliptic Modular Functions, An Introduction. Springer, Berlin (1974)CrossRefGoogle Scholar
  15. 15.
    Yui, N., Zagier, D.B.: On the singular values of Weber functions. Math. Comp. 66(220), 1645–1662 (1997)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Dept. of Mathematical SciencesIndiana University - Purdue University at Indianapolis (IUPUI)IndianapolisUSA

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