The Ramanujan Journal

, Volume 47, Issue 2, pp 417–425 | Cite as

On algebraic values of function \(\exp ~(2\pi i ~x+\log \log y)\)

  • Igor NikolaevEmail author


It is proved that, for all but a finite set of the square-free integers, d the value of transcendental function \(\exp ~(2\pi i ~x+\log \log y)\) is an algebraic number for the algebraic arguments x and y lying in a real quadratic field of discriminant, d. Such a value generates the Hilbert class field of the imaginary quadratic field of discriminant, \(-d\).


Real multiplication Sklyanin algebra Noncommutative tori 

Mathematics Subject Classification

11J81 (transcendence theory) 46L85 (noncommutative topology) 


  1. 1.
    Baker, A.: Transcendental Number Theory. Cambridge University Press, Cambridge (1975)CrossRefGoogle Scholar
  2. 2.
    Blackadar, B.: \(K\)-Theory for Operator Algebras. MSRI Publications, Springer, Berlin (1986)CrossRefGoogle Scholar
  3. 3.
    Feigin, B.L., Odesskii, A.V.: Sklyanin’s elliptic algebras. Funct. Anal. Appl. 23, 207–214 (1989)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Hilbert, D.: Mathematical problems. Bull. Am. Math. Soc. 8, 437–479 (1902)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Manin, Y.I.: Real multiplication and noncommutative geometry. In: Laudal, O.A., Piene, R. (eds.) Legacy of Niels Hendrik Abel, pp. 685–727. Springer, Berlin (2004)CrossRefGoogle Scholar
  6. 6.
    Murphy, G.J.: \(C^*\)-Algebras and Operator Theory. Academic Press, Cambridge (1990)zbMATHGoogle Scholar
  7. 7.
    Nikolaev, I.: On a symmetry of complex and real multiplication. Hokkaido Math. J. 45, 43–51 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Rieffel, M.A.: Non-commutative tori—a case study of non-commutative differentiable manifolds. Contemp. Math. 105, 191–211 (1990).
  9. 9.
    Smith, S.P., Stafford, J.T.: Regularity of the four dimensional Sklyanin algebra. Compos. Math. 83, 259–289 (1992)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Stafford, J.T., van den Bergh, M.: Noncommutative curves and noncommutative surfaces. Bull. Am. Math. Soc. 38, 171–216 (2001)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceSt. John’s UniversityNew YorkUSA

Personalised recommendations