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Singular moduli for a distinguished non-holomorphic modular function

Authors: Valerio Dose, Nathan Green, Michael Griffin, Tianyi Mao, Larry Rolen and John Willis
Journal: Proc. Amer. Math. Soc. 143 (2015), 965-972
MSC (2010): Primary 11F12, 11G15
Published electronically: October 29, 2014
MathSciNet review: 3293714
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Abstract: Here we study the integrality properties of singular moduli of a special non-holomorphic function $ \gamma (z)$, which was previously studied by Siegel, Masser, and Bruinier, Sutherland, and Ono. Similar to the modular $ j$-invariant, $ \gamma $ has algebraic values at any CM-point. We show that primes dividing the denominators of these values must have absolute value less than that of the discriminant and are not split in the corresponding quadratic field. Moreover, we give a bound for the size of the denominator.

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  • [1] A. Baker, On the periods of the Weierstrass ℘-function, Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69) Academic Press, London, 1970, pp. 155–174. MR 0279042
  • [2] A. Borel, S. Chowla, C. S. Herz, K. Iwasawa, and J.-P. Serre, Seminar on complex multiplication, Seminar held at the Institute for Advanced Study, Princeton, N.J., 1957-58. Lecture Notes in Mathematics, No. 21, Springer-Verlag, Berlin-New York, 1966. MR 0201394
  • [3] Jan Hendrik Bruinier, Andrew V. Sutherand, and Ken Ono, Class polynomials for nonholomorphic modular functions, preprint at
  • [4] David A. Cox, Primes of the form 𝑥²+𝑛𝑦², A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1989. Fermat, class field theory and complex multiplication. MR 1028322
  • [5] Benedict H. Gross and Don B. Zagier, On singular moduli, J. Reine Angew. Math. 355 (1985), 191–220. MR 772491
  • [6] Eric Larson and Larry Rolen, Integrality properties of the CM-values of certain weak Maass forms, to appear in Forum Math.
  • [7] Kristin Lauter and Bianca Viray, On singular moduli for arbitrary discriminants, preprint at
  • [8] David Masser, Elliptic functions and transcendence, Lecture Notes in Mathematics, Vol. 437, Springer-Verlag, Berlin-New York, 1975. MR 0379391
  • [9] Ken Ono, The web of modularity: arithmetic of the coefficients of modular forms and 𝑞-series, CBMS Regional Conference Series in Mathematics, vol. 102, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2004. MR 2020489
  • [10] Carl Ludwig Siegel, Bestimmung der elliptischen Modulfunktion durch eine Transformationsgleichung, Abh. Math. Sem. Univ. Hamburg 27 (1964), 32–38 (German). MR 0165102,
  • [11] Don Zagier, Elliptic modular forms and their applications, The 1-2-3 of modular forms, Universitext, Springer, Berlin, 2008, pp. 1–103. MR 2409678,

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Additional Information

Valerio Dose
Affiliation: Department of Mathematics, University of Rome Tor Vergata, Via della Ricerca Scientifica, 00133 Roma, Italy

Nathan Green
Affiliation: Department of Mathematics, 275 TMCB Brigham Young University, Provo, Utah 84602

Michael Griffin
Affiliation: Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322

Tianyi Mao
Affiliation: The Graduate Center, City University of New York, 365 Fifth Avenue, Room 4208, New York, New York 10016

Larry Rolen
Affiliation: Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322

John Willis
Affiliation: Department of Mathematics, University of South Carolina, 1523 Greene Street, Columbia, South Carolina 29208

Received by editor(s): March 25, 2013
Received by editor(s) in revised form: March 26, 2013, and June 19, 2013
Published electronically: October 29, 2014
Communicated by: Ken Ono
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.