Abstract
Duke and the second author defined a family of linear maps from spaces of weakly holomorphic modular forms of negative integral weight and level 1 into spaces of weakly holomorphic modular forms of half-integral weight and level 4 and showed that these lifts preserve the integrality of Fourier coefficients. We show that the generalization of these lifts to modular forms of genus 0 odd prime level also preserves the integrality of Fourier coefficients.
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References
Alfes, C.: Formulas for the coefficients of half-integral weight harmonic Maaß forms. Math. Z. 277(3–4), 769–795 (2014)
Atkin, A.O.L., Lehner, J.: Hecke operators on \(\Gamma _{0}(m)\). Math. Ann. 185, 134–160 (1970)
Basmaji, J.: Ein Algorithmus zur Berechnung von Hecke-Operatoren und Anwendungen auf modulare Kurven. PhD Dissertation, Universität Gesamthochschule Essen (1996). http://wstein.org/scans/papers/basmaji/
Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24, 235–365 (1997) (Computational Algebra and Number Theory (London (1993))
Bringmann, K., Ono, K.: Arithmetic properties of coefficients of half-integral weight Maass–Poincaré series. Math. Ann. 337(3), 591–612 (2007). http://www.mathcs.emory.edu/ono/publications-cv/pdfs/096
Bruinier, J.H., Jenkins, P., Ono, K.: Hilbert class polynomials and traces of singular moduli. Math. Ann. 334(2), 373–393 (2006). http://www.mathcs.emory.edu/ono/publications-cv/pdfs/090
Bruinier, J.H., van der Geer, G., Harder, G., Zagier, D.: Lectures from the summer school on modular forms and their applications held in Nordfjordeid, June 2004. In: Ranestad, K. (ed.) The 1-2-3 of Modular Forms. Universitext. Springer, Berlin (2008)
Duke, W., Jenkins, P.: Integral traces of singular values of weak Maass forms. Algebra Number Theory 2(5), 573–593 (2008)
El-Guindy, A.: Linear congruences and relations on spaces of cusp forms. Int. J. Number Theory 3(4), 529–539 (2007)
Garthwaite, S.A., Jenkins, P.: Zeros of weakly holomorphic modular forms of levels 2 and 3. Math. Res. Lett. 20(4), 657–674 (2013). arxiv.org/pdf/1205.7050v1
Kilford, L.J.P.: Modular Forms: A Classical and Computational Introduction. Imperial College Press, London (2008)
Kohnen, W.: Fourier coefficients of modular forms of half-integral weight. Math. Ann. 271(2), 237–268 (1985)
Miller, A., Pixton, A.: Arithmetic traces of non-holomorphic modular invariants. Int. J. Number Theory 6(1), 69–87 (2010)
Ono, K.: Unearthing the visions of a master: harmonic Maass forms and number theory. In: Current Developments in Mathematics, 2008, pp. 347–454. International Press, Somerville, MA (2009)
Rouse, J., Webb, J.J.: On spaces of modular forms spanned by eta-quotients. Adv. Math. 272, 200–224 (2015)
Serre, J.-P., Zagier, D.B. (eds.): Modular Functions of One Variable, VI. Lecture Notes in Mathematics, vol. 627. Springer, Berlin (1977)
Stein, W.: Modular Forms, A Computational Approach. Graduate Studies in Mathematics (With an Appendix by Paul E. Gunnells), vol. 79. American Mathematical Society, Providence (2007)
Stein, W., et al.: Sage Mathematics Software (Version 4.6). The Sage Development Team. http://www.sagemath.org (2012)
Ueda, M.: The decomposition of the spaces of cusp forms of half-integral weight and trace formula of Hecke operators. J. Math. Kyoto Univ. 28(3), 505–555 (1988)
Zagier, D.: Traces of singular moduli. In: Motives, Polylogarithms and Hodge Theory, Part I (Irvine, CA, 1998). International Press Lecture Series, vol. 3, pp. 211–244. International Press, Somerville, MA (2002). http://people.mpim-bonn.mpg.de/zagier/files/tex/TracesSingModuli/fulltext
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This work was partially supported by a grant from the Simons Foundation (#281876 to Paul Jenkins).
Appendix
Appendix
Here we list the beginning of the Fourier expansions of the form with the highest leading exponent in each of the bases described in Sect. 3 (Tables 1, 2, 3, and 4).
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Green, N., Jenkins, P. Integral traces of weak Maass forms of genus zero odd prime level. Ramanujan J 42, 453–478 (2017). https://doi.org/10.1007/s11139-015-9769-6
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DOI: https://doi.org/10.1007/s11139-015-9769-6