Singular moduli for a distinguished non-holomorphic modular function
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- by Valerio Dose, Nathan Green, Michael Griffin, Tianyi Mao, Larry Rolen and John Willis PDF
- Proc. Amer. Math. Soc. 143 (2015), 965-972 Request permission
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.References
- A. Baker, On the periods of the Weierstrass $\wp$-function, Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69) Academic Press, London, 1970, pp. 155–174. MR 0279042
- A. Borel, S. Chowla, C. S. Herz, K. Iwasawa, and J.-P. Serre, Seminar on complex multiplication, Lecture Notes in Mathematics, No. 21, Springer-Verlag, Berlin-New York, 1966. Seminar held at the Institute for Advanced Study, Princeton, N.J., 1957-58. MR 0201394
- Jan Hendrik Bruinier, Andrew V. Sutherand, and Ken Ono, Class polynomials for nonholomorphic modular functions, preprint at http://arxiv.org/abs/1301.5672.
- David A. Cox, Primes of the form $x^2 + ny^2$, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1989. Fermat, class field theory and complex multiplication. MR 1028322
- Benedict H. Gross and Don B. Zagier, On singular moduli, J. Reine Angew. Math. 355 (1985), 191–220. MR 772491
- Eric Larson and Larry Rolen, Integrality properties of the CM-values of certain weak Maass forms, to appear in Forum Math.
- Kristin Lauter and Bianca Viray, On singular moduli for arbitrary discriminants, preprint at http://arxiv.org/abs/1206.6942.
- David Masser, Elliptic functions and transcendence, Lecture Notes in Mathematics, Vol. 437, Springer-Verlag, Berlin-New York, 1975. MR 0379391
- 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
- Carl Ludwig Siegel, Bestimmung der elliptischen Modulfunktion durch eine Transformationsgleichung, Abh. Math. Sem. Univ. Hamburg 27 (1964), 32–38 (German). MR 165102, DOI 10.1007/BF02993054
- Don Zagier, Elliptic modular forms and their applications, The 1-2-3 of modular forms, Universitext, Springer, Berlin, 2008, pp. 1–103. MR 2409678, DOI 10.1007/978-3-540-74119-0_{1}
Additional Information
- Valerio Dose
- Affiliation: Department of Mathematics, University of Rome Tor Vergata, Via della Ricerca Scientifica, 00133 Roma, Italy
- Email: dose@mat.uniroma2.it
- Nathan Green
- Affiliation: Department of Mathematics, 275 TMCB Brigham Young University, Provo, Utah 84602
- Email: jaicouru@gmail.com
- Michael Griffin
- Affiliation: Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322
- Email: mjgrif3@emory.edu
- Tianyi Mao
- Affiliation: The Graduate Center, City University of New York, 365 Fifth Avenue, Room 4208, New York, New York 10016
- Email: tmao@gc.cuny.edu
- Larry Rolen
- Affiliation: Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322
- MR Author ID: 923990
- ORCID: 0000-0001-8671-8117
- Email: lrolen@mathcs.emory.edu
- John Willis
- Affiliation: Department of Mathematics, University of South Carolina, 1523 Greene Street, Columbia, South Carolina 29208
- Email: willisj5@mailbox.sc.edu
- 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
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 143 (2015), 965-972
- MSC (2010): Primary 11F12, 11G15
- DOI: https://doi.org/10.1090/S0002-9939-2014-12289-1
- MathSciNet review: 3293714