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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 | References | Similar Articles | Additional Information

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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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
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

DOI: https://doi.org/10.1090/S0002-9939-2014-12289-1
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.