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

  • Patrick MortonEmail author


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.


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

Mathematics Subject Classification

11D41 11G07 11G15 14H05 



  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

Personalised recommendations