The Ramanujan Journal

, Volume 41, Issue 1–3, pp 319–322 | Cite as

Transcendence of zeros of Jacobi forms

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Abstract

A special case of a fundamental theorem of Schneider asserts that if \(j(\tau )\) is algebraic (where j is the classical modular invariant), then any zero z not in \(\mathbf{Q}.L_\tau := \mathbf{Q}\oplus \mathbf{Q}\tau \) of the Weierstrass function \(\wp (\tau ,\cdot )\) attached to the lattice \(L_\tau =\mathbf{Z}\oplus \mathbf{Z}\tau \) is transcendental. In this note we generalize this result to holomorphic Jacobi forms of weight k and index \(m\in \mathbf{N}\) with algebraic Fourier coefficients.

Keywords

Jacobi forms Zeros Transcendency 

Mathematics Subject Classification

Primary 11F50 Secondary 11J81 

References

  1. 1.
    Berndt, R.: Zur Arithmetik der elliptischen Funktionenkörper höherer Stufe. J. Reine Angew. Math. 326, 79–94 (1981)MathSciNetMATHGoogle Scholar
  2. 2.
    Bruinier, J.H., Kohnen, W., Ono, K.: The arithmetic of the values of modular functions and the divisors of modular forms. Compos. Math. 140, 552–566 (2004)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Choie, Y., Kohnen, W.: Special values of elliptic functions at points of the divisors of Jacobi forms. Proc. Am. Math. Soc. 131(11), 3309–3317 (2003)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Duke, W., Imamoglu, Ö.: The zeros of the Weierstrass \(\wp \)-function and hypergeometric series. Math. Ann. 340, 897–905 (2008)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Eichler, M., Zagier, D.: The Theory of Jacobi Forms. Progress in Mathematics, vol. 55. Birkhäuser, Boston (1985)MATHGoogle Scholar
  6. 6.
    Gun, S., Ram Murty, M., Rath, P.: Algebraic independence of values of modular forms. Int. J. Number Theory 7(4), 1069–1074 (2011)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Schneider, T.: Arithmetische Untersuchungen elliptischer Integrale. Math. Ann. 113, 1–13 (1937)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of MathematicsPohang Institute of Science and Technology, POSTECHPohangKorea
  2. 2.Mathematisches InstitutUniversität HeidelbergHeidelbergGermany

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